Research ArticleSecond Atom-Bond Connectivity Index of Special ChemicalMolecular Structures

Wei Gao1 and Weifan Wang2

1 School of Information Science and Technology Yunnan Normal University Kunming 650500 China2Department of Mathematics Zhejiang Normal University Jinhua 321004 China

Correspondence should be addressed to Wei Gao gaoweiynnueducn

Received 7 July 2014 Accepted 27 August 2014 Published 15 October 2014

Academic Editor Demeter Tzeli

Copyright copy 2014 W Gao and W Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In theoretical chemistry the second atom-bond connectivity index was introduced tomeasure the stability of alkanes and the strainenergy of cycloalkanes In this paper we determine the second atom-bond connectivity index of unilateral polyomino chain andunilateral hexagonal chain Furthermore the second ABC indices of V-phenylenic nanotubes and nanotori are presented

1 Introduction

One of the most important applications of chemical graphtheory is to measure chemical physical and pharmaceuticalproperties of molecules called alkanes Several indices thatrelied on the graphical structure of the alkanes are definedand employed to model both the melting point and boilingpoint of the molecules Molecular graph is a topologicalrepresentation of a molecule such that each vertex representsan atom of molecule and covalent bonds between atoms arerepresented by edges between the corresponding vertices

Specifically topological index can be regarded as a scorefunction119891 119866 rarr R+ with this property that119891(119866

1) = 119891(119866

2)

if two molecular graphs 1198661and 119866

2are isomorphic There are

several vertex distance-based anddegree-based indiceswhichare introduced to analyze the chemical properties ofmoleculegraph for instanceWiener index PI index Szeged index andatom-bond connectivity index Several papers contributed todetermining the indices of special molecular graphs (see Yanet al [1 2] Gao and Shi [3] Xi and Gao [4] and Dou et al [5]for more detail)

All (molecular) graphs considered in this paper are finiteloopless and without multiple edges Let 119866 be a (molecular)graph with vertex set 119881(119866) and edge set 119864(119866) All graphnotations and terminologies used but undefined in this papercan be found in [6]

The atom-bond connectivity index (shortly 119860119861119862 index)is defined by Estrada et al [7] as

119860119861119862 (119866) = sum119906Visin119864(119866)

radic119889 (119906) + 119889 (V) minus 2

119889 (119906) 119889 (V) (1)

Das et al [8] characterized the molecular graphs as extremalwith regard to 119860119861119862 index Furtula et al [9] found the chem-ical trees with extremal119860119861119862 index Vassilev and Huntington[10] identified certain classes of edges that are importantand occur frequently in chemical trees and continuouslystudied the chemical trees with extremal119860119861119862 index in termsof learning how the removal of a certain edge takes placeChen et al [11] obtained the atom-bond connectivity index ofthe zig-zag chain polyominomolecular graphs Furthermorethey determined the sharp upper bound on the atom-bondconnectivity index of catacon densed polyomino moleculargraphs with squares and determined the correspondingextremal molecular graphs

Recently Graovac and Ghorbani [12] defined a newversion of the atom-bond connectivity index that is thesecond atom-bond connectivity index (shortly second 119860119861119862index)

1198601198611198622(119866) = sum

119906Visin119864(119866)

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V) (2)

Hindawi Publishing CorporationJournal of ChemistryVolume 2014 Article ID 906254 8 pageshttpdxdoiorg1011552014906254

2 Journal of Chemistry

Linear polyomino chain Zig-zag polyomino chain

middot middot middot

Figure 1 The structure of 1198714119899and 1198854

119899

where for each 119906V isin 119864(119866) 119899(119906) is the number of verticescloser to vertex 119906 than vertex V and 119899(V) defines similarlyIn the chemistry applications second 119860119861119862 index is usedto model both the boiling point and melting point of themolecules Hence it is also applied to the pharmaceuticalfield Rostami et al [13] calculated some upper bounds forthe second atom-bond connectivity index

Although there have been several advances in atom-bondconnectivity index of molecular graphs the study of secondatom-bond connectivity index of special chemical structureshas been largely limited In addition as widespread andcritical chemical structures polyomino system hexagonalsystem V-phenylenic nanotubes and nanotori are widelyused in medical science and pharmaceutical field As anexample polyomino chain is one of the basic chemicalstructures and exists widely in benzene and alkali molecularstructures For these reasons we have attracted tremendousacademic and industrial interests to research the secondatom-bond connectivity of these molecular structures froma mathematical point of view

The contribution of our paper is threefold First we com-pute the second atom-bond connectivity index of unilateralpolyomino chain Then the second atom-bond connectivityindex of unilateral hexagonal chain is calculated At lastwe derive the second atom-bond connectivity index of V-phenylenic nanotubes and nanotori

2 Second Atom-Bond Connectivity Index ofPolyomino Chain

From the view of graph theory polymino is a finite 2-connected planar graph and each interior face is surroundedby a square with length 4 Polyomino chain is one class ofpolyomino such that the connection of centres for adjacentsquares constitutes a path 119888

11198882sdot sdot sdot 119888119899 where 119888

119894is the centre of

119894th square Polyomino chain 1198674119899is called a linear chain if the

subgraph induced by all 3-degree vertices is a graphwith 119899 minus 2

squares Furthermore polyomino chain 1198674119899is called a zig-

zag chain if the subgraph induced by all vertices with degreegt 2 is path with 119899 minus 1 edges In what follows we use 1198714

119899and

1198854119899to denote linear polyomino chain and zig-zag polyomino

chain respectively For the structure of 1198714119899and 1198854

119899 refer to

Figure 1Use the similar technology raised in Gutman and Klavzar

[14] and we define elementary cut as follows Choose an edge

119890 of the polyomino system and draw a straight line throughthe center of 119890 orthogonal on 119890 This line will intersect theperimeter in two end points 119875

1and 119875

2 The straight line

segment 119862 whose end points are 1198751and 119875

2is the elementary

cut intersecting the edge 119890 A fragment 119878 in polyomino chainis just maximal linear chain which includes the squares instart and end vertices Let 119897(119878) be the length of fragmentwhich denotes the number of squares it contained Let 1198674

119899

be a polyomino chain with 119899 squares consisting of fragmentsequence 119878

1 1198782 119878

119898(119898 ge 1) Denote 119897(119878

119894) = 119897

119894(119894 =

1 119898) It is not difficult to verify that 1198971

+ 1198972

+ sdot sdot sdot + 119897119898

=

119899 + 119898 minus 1 and |119881(1198674119899)| = 2119899 + 2 |119864(1198674

119899)| = 3119899 + 1 For the

119896th fragment of polyomino chain the cut of this fragment isthe cut which intersects with 119897

119896+ 1 parallel edges of squares

in this fragment A fragment is called horizontal fragment ifits cut parallels the horizontal direction and called verticalfragment if its cut parallels the vertical direction Unilateralpolyomino chain is a special kind of polyomino chain suchthat for each vertical fragment two horizontal fragments (ifexists) are adjacent and it appears in the left and right sidesrespectively

Our main result in this section stated as follows presentsthe second atom-bond connectivity index of unilateral poly-omino chain

Theorem 1 Let1198674119899be a unilateral polyomino chain consisting

of 119898 fragment 1198781 1198782 119878

119898(119898 ge 1) and let 119897(119878

119894) = 119897119894(119894 =

1 119898) be the length of each fragment Then one has

1198601198611198622(1198674119899)

= 21198971minus1

sum119895=1

radic119899

119895 (2sum119898

119894=2119897119894minus 2119898 + 2 (119897

1minus 119895) + 4)

+ 2119897119898

sum119895=2

radic119899

(2sum119898minus1

119894=1119897119894minus 2119898 + 2119895 + 2) (119897

119898minus 119895 + 1)

+ 2119898minus1

sum119896=2

119897119896minus1

sum119895=2

(119899((2119896minus1

sum119894=1

119897119894minus 2119896 + 2119895 + 2)

times (119898

sum119894=119896+1

119897119894minus (119898 minus 119896) + (119897

119896minus 119895) + 1))

minus1

)

12

Journal of Chemistry 3

II III

I CP1 P2

II II I

I

I

II II

middot middot middot

middot middot middot

middot middot middot

Figure 2 I-type cut and II-type cut of unilateral polyomino chain

+119898

sum119896=1

(119897119896+ 1)

times (2119899((2119896minus1

sum119894=1

119897119894minus 2119896 + 119897

119896+ 3)

times (2119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 119897

119896+ 1))

minus1

)

12

(3)

Proof The cuts in 1198674119899are divided into two types I-type and

II-type (see Figure 2) An edge is called I-type (or II-type) ifit intersects with I-type (or II-type) cut Now we consider thefollowing two cases

Case 1 If edge 119890 is I-type in 119895th square of 119896th fragment (ie 119890is edge which is passed by dotted line in Figure 2) we observethat there is 119897

119896+ 1 such edges in 119896th fragment

Subcase 11 If 119896 = 1 then we have

1198991(119890) = 119897

1+ 1

1198992(119890) = 2

119898

sum119894=2

119897119894minus 2119898 + 119897

1+ 3

(4)

Subcase 12 If 119896 = 119898 then we obtain

1198991(119890) = 2

119898minus1

sum119894=1

119897119894minus 2119898 + 119897

119898+ 3

1198992(119890) = 119897

119898+ 1

(5)

Subcase 13 If 2 le 119896 le 119898 minus 1 then we get

1198991(119890) = 2

119896minus1

sum119894=1

119897119894minus 2119896 + 119897

119896+ 3

1198992(119890) = 2

119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 119897

119896+ 1

(6)

Case 2 If edge 119890 is II-type in 119895th square of 119896th fragment (ie119890 is the edge which is passed by real line in Figure 2)

Subcase 21 If 119896 = 1 then we yield

1198991(119890) = 2119895

1198992(119890) = 2

119898

sum119894=2

119897119894minus 2119898 + 2 (119897

1minus 119895) + 4

(7)

Subcase 22 If 119896 = 119898 then we infer

1198991(119890) = 2

119898minus1

sum119894=1

119897119894minus 2119898 + 2119895 + 2

1198992(119890) = 2119897

119898minus 2119895 + 2

(8)

Subcase 23 If 2 le 119896 le 119898 minus 1 then we deduce

1198991(119890) = 2

119896minus1

sum119894=1

119897119894minus 2119896 + 2119895 + 2

1198992(119890) = 2

119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 2 (119897

119896minus 119895) + 2

(9)

Hence the desired conclusion is obtained by combiningthe above case and the definition of the second atom-bondconnectivity index

Corollary 2 Let 1198714119899be the linear chain with 119899 squares Then

one has

1198601198611198622(1198714119899) = 2119899minus1

sum119895=1

radic119899

2119895 (119899 minus 119895 + 1)+ radic2119899 (10)

Proof By the definition of linear chain we have 119898 = 1 1198971=

119899 1198972= sdot sdot sdot = 119897

119898= 0 In terms of Theorem 1 we immediately

get the result

Corollary 3 Let1198854119899be the zig-zag chain with 119899 squaresThen

one has

1198601198611198622(1198854119899) = 4radic

1

2+119898

sum119896=1

3radic2119899

(2119896 + 1) (2119899 minus 2119896 + 1) (11)

Proof By the virtue of the definition of the zig-zag chain wehave 119898 = 119899 minus 1 and 119897

1= 1198972

= sdot sdot sdot = 119897119898

= 2 In view ofTheorem 1 the result is immediately obtained

4 Journal of Chemistry

Linear hexagonal chain Zig-zag hexagonal chain

middot middot middot

middot middot middot

middot middot middot

Figure 3 The structure of 1198716119899and 1198856

119899

3 Second Atom-Bond Connectivity Index ofHexagonal Chain

Hexagonal chain is one class of hexagonal system consistingof hexagonal In hexagonal chain each of two hexagonal hasone common edge or no common vertex Two hexagonalare adjacent if they have common edge No three or morehexagonal share one vertex Each hexagonal has two adjacenthexagonal except hexagonal in terminus and each hexagonalchain has two hexagonal in terminus

It is easy to verify that the hexagonal chain with119899 hexagonal has 4119899 + 2 vertices and 5119899 + 1 edges Let 1198716

119899and

1198856119899be the linear hexagonal chain and zig-zag hexagonal chain

respectively For the chemical structure of 1198716119899and1198856

119899 one can

refer to Figure 3 for more detailAgain we use the similar trick which was presented in

Gutman and Klavzar [14] and we define elementary cut asfollows Choose an edge 119890 of the hexagonal system and drawa straight line through the center of 119890 orthogonal on 119890 Thisline will intersect the perimeter in two end points 119875

1and 119875

2

The straight line segment 119862 whose end points are 1198751and 119875

2

is the elementary cut intersecting the edge 119890 A fragment 119878 inhexagonal chain is just maximal linear chain which includesthe hexagonal in start and end vertices Let 1198971015840(119878) be thelength of fragment which denotes the number of hexagonalcontained Let 1198676

119899be a hexagonal chain with 119899 hexagonal

consisting of fragment sequence 1198781 1198782 119878

119898(119898 ge 1)

Denote 1198971015840(119878119894) = 1198971015840119894(119894 = 1 119898) Then we verify that 1198971015840

1+ 11989710158402+

sdot sdot sdot + 1198971015840119898

= 119899 + 119898 minus 1 since each two adjacent fragments haveone common hexagonal For the 119896th fragment of hexagonalchain the cut of this fragment is the cut which intersectswith 1198971015840

119896+ 1 parallel edges of hexagonal in this fragment A

fragment is called horizontal fragment if its cut parallels thehorizontal direction otherwise it is called inclined fragmentUnilateral hexagonal chain is a special class of hexagonalchain such that the cut for each inclined fragment at the sameangle with a horizontal direction As an example Figure 4shows a structure of unilateral hexagonal chain Clearlylinear hexagonal chain 1198716

119899is a unilateral hexagonal chain with

one fragment and zig-zag is a unilateral hexagonal chainwith119899 minus 1 fragments

I II

IC

II

II

IIII

II

II

II

II

II I

I

I

P1 P2

middot middot middot

middot middot middot

⋱

Figure 4 I-type cut and II-type cut of unilateral hexagonal chain

We now show the second atom-bond connectivity indexof unilateral hexagonal chain

Theorem 4 Let1198676119899be a unilateral hexagonal chain consisting

of 119898 fragment 1198781 1198782 119878

119898(119898 ge 1) and let 1198971015840(119878

119894) = 1198971015840119894(119894 =

1 119898) be the length of each fragment Then one has

1198601198611198622(1198676119899)

= 4

1198971015840

1minus1

sum119895=1

radic4119899

(4119895 minus 1) [4 (sum119898

119894=21198971015840119894minus 119898 + 1) + 41198971015840

1minus 4119895 + 3]

+ 2119898minus1

sum119896=1

(4119899([4(119896

sum119894=1

1198971015840119894minus 119896 + 1) minus 1]

times[4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 3])

minus1

)

12

+ 4119898minus1

sum119896=2

1198971015840

119896minus1

sum119895=2

(4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 4119895 minus 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896)

+41198971015840119896minus 4119895 + 3])

minus1

)

12

Journal of Chemistry 5

+ 4

1198971015840

119898

sum119895=2

(4119899([4(119898minus1

sum119894=1

1198971015840119894minus 119898 + 1) + 4119895 minus 1]

times(41198971015840119898

minus 4119895 + 3))

minus1

)

12

+119898minus1

sum119896=2

(1198971015840119896+ 1)

times (4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1])

minus1

)

12

(12)

Proof The cuts in 1198676119899are divided into two types I-type and

II-type (see Figure 4) An edge is called I-type if it intersectswith I-type cut Also an edge is called II-type if it intersectswith II-type cut In what follows we consider two situationsaccording to the type of an edge

Case 1 If edge 119890 is I-type in 119895th square of 119896th fragment (ie 119890is edge which is passed by dotted line in Figure 4)

Subcase 11 If 119896 = 1 then we have

1198991(119890) = 21198971015840

1+ 1

1198992(119890) = 4

119898

sum119894=2

(1198971015840119894minus 119898 + 1) + 21198971015840

1+ 1

(13)

Subcase 12 If 119896 = 119898 then we obtain

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119898 + 1) + 21198971015840

119896+ 1

1198992(119890) = 21198971015840

119898+ 1

(14)

Subcase 13 If 2 le 119896 le 119898 minus 1 then we get

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1

1198992(119890) = 4

119898

sum119894=119896+1

(1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1

(15)

Case 2 If edge 119890 is II-type in 119895th square of 119896th fragment (ie119890 is the edge which is passed by real line in Figure 4)

Subcase 21 If 119896 = 1 then we yield

1198991(119890) = 4119895 minus 1

1198992(119890) = 4

119898

sum119894=2

1198971015840119894+ 411989710158401minus 4119895 + 3

(16)

Subcase 22 If 119896 = 119898 then we infer

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 41198971015840

119898minus 4119895 + 3

(17)

Subcase 23 If 2 le 119896 le 119898 minus 1 then we deduce

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 4

119898

sum119894=119896+1

1198971015840119894+ 4 (1198971015840119896minus 119895) + 3

(18)

In particular if 119895 = 1198971015840119896in Subcase 23 we have 119899

1(119890) =

4sum119896

119894=1(1198971015840119894minus 119896 + 1) minus 1 and 119899

2(119890) = 4sum

119898

119894=119896+1(1198971015840119894minus 119898 + 119896) + 3

Hence we get the desired result by combining the abovecases and the definition of the second atom-bond connectiv-ity index

Corollary 5 Let 1198716119899be the linear chain with 119899 hexagonal

Then one has

1198601198611198622(1198716119899) = 4119899minus1

sum119895=1

radic4119899

(4119895 minus 1) (4119899 minus 4119895 + 3) (19)

Proof By the definition of linear chain we check that 119898 =1 11989710158401= 119899 11989710158402= sdot sdot sdot = 1198971015840

119898= 0 In terms ofTheorem 4 we shortly

get the result

Corollary 6 Let 1198856119899be the zig-zag chain with 119899 hexagonal

Then one has

1198601198611198622(1198856119899) = 8radic

4119899

3 (4119899 minus 1)

+ 2119899minus2

sum119896=1

radic4119899

(4119896 + 3) (4119899 + 4119896 minus 1)

+ 3119899minus2

sum119896=2

radic4119899

(4119896 + 1) (4119899 minus 4119896 + 1)

(20)

Proof Using the definition of zig-zag chain we verify that119898 = 119899 minus 1 and 1198971015840

1= 11989710158402= sdot sdot sdot = 1198971015840

119898= 2 In view of Theorem 4

the corollary is immediately yielded

4 Second Atom-Bond Connectivity Index ofV-Phenylenic Nanotubes and Nanotori

The notations in this section follow from Diudea [15]In what follows we use 119899(119906) and 119899(V) to express 119899

1(119890)

and 1198992(119890) for short respectively The molecular structures

V-phenylenic nanotube and V-phenylenic nanotorus aredenoted by 119881119875119867119883[119898 119899] and 119881119875119867119884[119898 119899] respectively

The next result shows the representation of1198601198611198622(119881119875119867119883[119898 119899]) and the structure of 119881119875119867119883[119898 119899]

is described in Figure 5

6 Journal of Chemistry

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 5 The structure of V-phenylenic nanotube

Theorem 7 Consider

1198601198611198622(119881119875119867119883 [119898 119899]) =

1205721 if 119898 = 119899

1205722 if 119898 = 119899

(21)

where 1205721and 120572

2are presented in the process of the proof

Proof First we note that |119881(119881119875119867119883[119898 119899])| = 6119898119899To compute the second atom-bond connectivity index of119881119875119867119883[119898 119899] we assume that 119864

1 1198642 and 119864

3are the set of all

vertical oblique and horizontal edges respectivelyThen weinfer

1198601198611198622(119881119875119867119883 [119898 119899])

= sum119890isin119864(119881119875119867119883[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(22)

Now we calculate sum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) by con-sidering the following two cases

Case 1 (119898 = 119899) Under this situation we deduce

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(23)where 119878

119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

in this case we infer that the second atom-bond connectivityindex of 119881119875119867119883[119898 119899] is

1205721= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(24)Case 2 (119898 = 119899) In this case we get

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(25)

Thus the corresponding second atom-bond connectivityindex is

1205722= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(26)The result is archived

Journal of Chemistry 7

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 6 The structure of V-phenylenic nanotorus

Our last result computes the second atom-bond connec-tivity index of 119881119875119867119884[119898 119899] and the structure of 119881119875119867119884[119898 119899]presented in Figure 6

Theorem 8 Consider

1198601198611198622(119881119875119867119884 [119898 119899]) =

1205723 if 119898 = 119899

1205724 if 119898 = 119899

(27)

where 1205723and 120572

4are presented in the process of the proof

Proof In order to prove the theorem we use a similar trickas in Theorem 7 Obviously |119881(119881119875119867119884[119898 119899])| = 6119898119899 Weassume that 119864

1 1198642 and 119864

3are the set of all vertical oblique

and horizontal edges respectively Then we deduce

1198601198611198622(119881119875119867119884 [119898 119899])

= sum119890isin119864(119881119875119867119884[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(28)

Now we calculatesum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) accord-ing to the relationship between 119898 and 119899

Case 1 (119898 = 119899) Under this assumption we infer

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ( (2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3)119894))

minus1

)

12

(29)

where 119878119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

the second atom-bond connectivity index of119881119875119867119884[119898 119899] for119898 = 119899 is

1205723= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894))

minus1

)12

(30)

Case 2 (119898 = 119899) In this case we have

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(31)

8 Journal of Chemistry

Thus the corresponding second atom-bond connectivityindex is

1205724= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(32)

The proof is completed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for their constructive com-ments in improving the quality of this paperThey also wouldlike to thank the anonymous referees for providing themwith constructive comments and suggestions The researchis partially supported by NSFC (nos 11401519 11371328 and11471293)

References

[1] L Yan Y Li W Gao and J Li ldquoPI index for some specialgraphsrdquo Journal of Chemical and Pharmaceutical Research vol5 no 11 pp 260ndash264 2013

[2] L Yan Y Li W Gao and J Li ldquoOn the extremal hyper-wienerindex of graphsrdquo Journal of Chemical and PharmaceuticalResearch vol 6 no 3 pp 477ndash481 2014

[3] W Gao and L Shi ldquoWiener index of gear fan graph and gearwheel graphrdquo Asian Journal of Chemistry vol 26 no 11 pp3397ndash3400 2014

[4] W F Xi and W Gao ldquoGeometric-arithmetic index and Zagrebindices of certain special molecular graphsrdquo Journal of Advancesin Chemistry vol 10 no 2 pp 2254ndash2261 2014

[5] J Y Dou Y Y Wang and W Gao ldquoSome characteristicson hyper-wiener index of graphsrdquo Journal of Chemical andPharmaceutical Research vol 6 no 5 pp 1659ndash1663 2014

[6] J A Bondy and U S R Murty GraphTheory Springer Berlin Germany 2008

[7] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry A vol 37 no 10 pp849ndash855 1998

[8] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquoChemical Physics Letters vol 511 pp 452ndash454 2011

[9] B Furtula A Graovac and D Vukicevic ldquoAtom-bond connec-tivity index of treesrdquo Discrete Applied Mathematics vol 157 no13 pp 2828ndash2835 2009

[10] T S Vassilev and L J Huntington ldquoOn the minimum ABCindex of chemical treesrdquo Applied Mathematics vol 2 no 1 pp8ndash16 2012

[11] J Chen J Liu and Q Li ldquoThe atom-bond connectivity index ofcatacondensed polyomino graphsrdquoDiscrete Dynamics inNatureand Society vol 2013 Article ID 598517 7 pages 2013

[12] A Graovac and M Ghorbani ldquoA new version of atom-bondconnectivity indexrdquo Acta Chimica Slovenica vol 57 no 3 pp609ndash612 2010

[13] M Rostami M S Haghighat and M Ghorbani ldquoOn secondatom-bond connectivity indexrdquo Iranian Journal of Mathemati-cal Chemistry vol 4 no 2 pp 265ndash270 2013

[14] I Gutman and S Klavzar ldquoAn algorithm for the calculationof the Szeged index of benzenoid hydrocarbonsrdquo Journal ofChemical Information and Computer Sciences vol 35 no 6 pp1011ndash1014 1995

[15] M V Diudea ldquoPhenylenic and naphthylenic torirdquo FullerenesNanotubes and Carbon Nanostructures vol 10 no 4 pp 273ndash292 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

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Quantum Chemistry

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CatalystsJournal of

2 Journal of Chemistry

Linear polyomino chain Zig-zag polyomino chain

middot middot middot

Figure 1 The structure of 1198714119899and 1198854

119899

where for each 119906V isin 119864(119866) 119899(119906) is the number of verticescloser to vertex 119906 than vertex V and 119899(V) defines similarlyIn the chemistry applications second 119860119861119862 index is usedto model both the boiling point and melting point of themolecules Hence it is also applied to the pharmaceuticalfield Rostami et al [13] calculated some upper bounds forthe second atom-bond connectivity index

Although there have been several advances in atom-bondconnectivity index of molecular graphs the study of secondatom-bond connectivity index of special chemical structureshas been largely limited In addition as widespread andcritical chemical structures polyomino system hexagonalsystem V-phenylenic nanotubes and nanotori are widelyused in medical science and pharmaceutical field As anexample polyomino chain is one of the basic chemicalstructures and exists widely in benzene and alkali molecularstructures For these reasons we have attracted tremendousacademic and industrial interests to research the secondatom-bond connectivity of these molecular structures froma mathematical point of view

The contribution of our paper is threefold First we com-pute the second atom-bond connectivity index of unilateralpolyomino chain Then the second atom-bond connectivityindex of unilateral hexagonal chain is calculated At lastwe derive the second atom-bond connectivity index of V-phenylenic nanotubes and nanotori

2 Second Atom-Bond Connectivity Index ofPolyomino Chain

From the view of graph theory polymino is a finite 2-connected planar graph and each interior face is surroundedby a square with length 4 Polyomino chain is one class ofpolyomino such that the connection of centres for adjacentsquares constitutes a path 119888

11198882sdot sdot sdot 119888119899 where 119888

119894is the centre of

119894th square Polyomino chain 1198674119899is called a linear chain if the

subgraph induced by all 3-degree vertices is a graphwith 119899 minus 2

squares Furthermore polyomino chain 1198674119899is called a zig-

zag chain if the subgraph induced by all vertices with degreegt 2 is path with 119899 minus 1 edges In what follows we use 1198714

119899and

1198854119899to denote linear polyomino chain and zig-zag polyomino

chain respectively For the structure of 1198714119899and 1198854

119899 refer to

Figure 1Use the similar technology raised in Gutman and Klavzar

[14] and we define elementary cut as follows Choose an edge

119890 of the polyomino system and draw a straight line throughthe center of 119890 orthogonal on 119890 This line will intersect theperimeter in two end points 119875

1and 119875

2 The straight line

segment 119862 whose end points are 1198751and 119875

2is the elementary

cut intersecting the edge 119890 A fragment 119878 in polyomino chainis just maximal linear chain which includes the squares instart and end vertices Let 119897(119878) be the length of fragmentwhich denotes the number of squares it contained Let 1198674

119899

be a polyomino chain with 119899 squares consisting of fragmentsequence 119878

1 1198782 119878

119898(119898 ge 1) Denote 119897(119878

119894) = 119897

119894(119894 =

1 119898) It is not difficult to verify that 1198971

+ 1198972

+ sdot sdot sdot + 119897119898

=

119899 + 119898 minus 1 and |119881(1198674119899)| = 2119899 + 2 |119864(1198674

119899)| = 3119899 + 1 For the

119896th fragment of polyomino chain the cut of this fragment isthe cut which intersects with 119897

119896+ 1 parallel edges of squares

in this fragment A fragment is called horizontal fragment ifits cut parallels the horizontal direction and called verticalfragment if its cut parallels the vertical direction Unilateralpolyomino chain is a special kind of polyomino chain suchthat for each vertical fragment two horizontal fragments (ifexists) are adjacent and it appears in the left and right sidesrespectively

Our main result in this section stated as follows presentsthe second atom-bond connectivity index of unilateral poly-omino chain

Theorem 1 Let1198674119899be a unilateral polyomino chain consisting

of 119898 fragment 1198781 1198782 119878

119898(119898 ge 1) and let 119897(119878

119894) = 119897119894(119894 =

1 119898) be the length of each fragment Then one has

1198601198611198622(1198674119899)

= 21198971minus1

sum119895=1

radic119899

119895 (2sum119898

119894=2119897119894minus 2119898 + 2 (119897

1minus 119895) + 4)

+ 2119897119898

sum119895=2

radic119899

(2sum119898minus1

119894=1119897119894minus 2119898 + 2119895 + 2) (119897

119898minus 119895 + 1)

+ 2119898minus1

sum119896=2

119897119896minus1

sum119895=2

(119899((2119896minus1

sum119894=1

119897119894minus 2119896 + 2119895 + 2)

times (119898

sum119894=119896+1

119897119894minus (119898 minus 119896) + (119897

119896minus 119895) + 1))

minus1

)

12

Journal of Chemistry 3

II III

I CP1 P2

II II I

I

I

II II

middot middot middot

middot middot middot

middot middot middot

Figure 2 I-type cut and II-type cut of unilateral polyomino chain

+119898

sum119896=1

(119897119896+ 1)

times (2119899((2119896minus1

sum119894=1

119897119894minus 2119896 + 119897

119896+ 3)

times (2119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 119897

119896+ 1))

minus1

)

12

(3)

Proof The cuts in 1198674119899are divided into two types I-type and

II-type (see Figure 2) An edge is called I-type (or II-type) ifit intersects with I-type (or II-type) cut Now we consider thefollowing two cases

Case 1 If edge 119890 is I-type in 119895th square of 119896th fragment (ie 119890is edge which is passed by dotted line in Figure 2) we observethat there is 119897

119896+ 1 such edges in 119896th fragment

Subcase 11 If 119896 = 1 then we have

1198991(119890) = 119897

1+ 1

1198992(119890) = 2

119898

sum119894=2

119897119894minus 2119898 + 119897

1+ 3

(4)

Subcase 12 If 119896 = 119898 then we obtain

1198991(119890) = 2

119898minus1

sum119894=1

119897119894minus 2119898 + 119897

119898+ 3

1198992(119890) = 119897

119898+ 1

(5)

Subcase 13 If 2 le 119896 le 119898 minus 1 then we get

1198991(119890) = 2

119896minus1

sum119894=1

119897119894minus 2119896 + 119897

119896+ 3

1198992(119890) = 2

119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 119897

119896+ 1

(6)

Case 2 If edge 119890 is II-type in 119895th square of 119896th fragment (ie119890 is the edge which is passed by real line in Figure 2)

Subcase 21 If 119896 = 1 then we yield

1198991(119890) = 2119895

1198992(119890) = 2

119898

sum119894=2

119897119894minus 2119898 + 2 (119897

1minus 119895) + 4

(7)

Subcase 22 If 119896 = 119898 then we infer

1198991(119890) = 2

119898minus1

sum119894=1

119897119894minus 2119898 + 2119895 + 2

1198992(119890) = 2119897

119898minus 2119895 + 2

(8)

Subcase 23 If 2 le 119896 le 119898 minus 1 then we deduce

1198991(119890) = 2

119896minus1

sum119894=1

119897119894minus 2119896 + 2119895 + 2

1198992(119890) = 2

119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 2 (119897

119896minus 119895) + 2

(9)

Hence the desired conclusion is obtained by combiningthe above case and the definition of the second atom-bondconnectivity index

Corollary 2 Let 1198714119899be the linear chain with 119899 squares Then

one has

1198601198611198622(1198714119899) = 2119899minus1

sum119895=1

radic119899

2119895 (119899 minus 119895 + 1)+ radic2119899 (10)

Proof By the definition of linear chain we have 119898 = 1 1198971=

119899 1198972= sdot sdot sdot = 119897

119898= 0 In terms of Theorem 1 we immediately

get the result

Corollary 3 Let1198854119899be the zig-zag chain with 119899 squaresThen

one has

1198601198611198622(1198854119899) = 4radic

1

2+119898

sum119896=1

3radic2119899

(2119896 + 1) (2119899 minus 2119896 + 1) (11)

Proof By the virtue of the definition of the zig-zag chain wehave 119898 = 119899 minus 1 and 119897

1= 1198972

= sdot sdot sdot = 119897119898

= 2 In view ofTheorem 1 the result is immediately obtained

4 Journal of Chemistry

Linear hexagonal chain Zig-zag hexagonal chain

middot middot middot

middot middot middot

middot middot middot

Figure 3 The structure of 1198716119899and 1198856

119899

3 Second Atom-Bond Connectivity Index ofHexagonal Chain

Hexagonal chain is one class of hexagonal system consistingof hexagonal In hexagonal chain each of two hexagonal hasone common edge or no common vertex Two hexagonalare adjacent if they have common edge No three or morehexagonal share one vertex Each hexagonal has two adjacenthexagonal except hexagonal in terminus and each hexagonalchain has two hexagonal in terminus

It is easy to verify that the hexagonal chain with119899 hexagonal has 4119899 + 2 vertices and 5119899 + 1 edges Let 1198716

119899and

1198856119899be the linear hexagonal chain and zig-zag hexagonal chain

respectively For the chemical structure of 1198716119899and1198856

119899 one can

refer to Figure 3 for more detailAgain we use the similar trick which was presented in

Gutman and Klavzar [14] and we define elementary cut asfollows Choose an edge 119890 of the hexagonal system and drawa straight line through the center of 119890 orthogonal on 119890 Thisline will intersect the perimeter in two end points 119875

1and 119875

2

The straight line segment 119862 whose end points are 1198751and 119875

2

is the elementary cut intersecting the edge 119890 A fragment 119878 inhexagonal chain is just maximal linear chain which includesthe hexagonal in start and end vertices Let 1198971015840(119878) be thelength of fragment which denotes the number of hexagonalcontained Let 1198676

119899be a hexagonal chain with 119899 hexagonal

consisting of fragment sequence 1198781 1198782 119878

119898(119898 ge 1)

Denote 1198971015840(119878119894) = 1198971015840119894(119894 = 1 119898) Then we verify that 1198971015840

1+ 11989710158402+

sdot sdot sdot + 1198971015840119898

= 119899 + 119898 minus 1 since each two adjacent fragments haveone common hexagonal For the 119896th fragment of hexagonalchain the cut of this fragment is the cut which intersectswith 1198971015840

119896+ 1 parallel edges of hexagonal in this fragment A

fragment is called horizontal fragment if its cut parallels thehorizontal direction otherwise it is called inclined fragmentUnilateral hexagonal chain is a special class of hexagonalchain such that the cut for each inclined fragment at the sameangle with a horizontal direction As an example Figure 4shows a structure of unilateral hexagonal chain Clearlylinear hexagonal chain 1198716

119899is a unilateral hexagonal chain with

one fragment and zig-zag is a unilateral hexagonal chainwith119899 minus 1 fragments

I II

IC

II

II

IIII

II

II

II

II

II I

I

I

P1 P2

middot middot middot

middot middot middot

⋱

Figure 4 I-type cut and II-type cut of unilateral hexagonal chain

We now show the second atom-bond connectivity indexof unilateral hexagonal chain

Theorem 4 Let1198676119899be a unilateral hexagonal chain consisting

of 119898 fragment 1198781 1198782 119878

119898(119898 ge 1) and let 1198971015840(119878

119894) = 1198971015840119894(119894 =

1 119898) be the length of each fragment Then one has

1198601198611198622(1198676119899)

= 4

1198971015840

1minus1

sum119895=1

radic4119899

(4119895 minus 1) [4 (sum119898

119894=21198971015840119894minus 119898 + 1) + 41198971015840

1minus 4119895 + 3]

+ 2119898minus1

sum119896=1

(4119899([4(119896

sum119894=1

1198971015840119894minus 119896 + 1) minus 1]

times[4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 3])

minus1

)

12

+ 4119898minus1

sum119896=2

1198971015840

119896minus1

sum119895=2

(4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 4119895 minus 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896)

+41198971015840119896minus 4119895 + 3])

minus1

)

12

Journal of Chemistry 5

+ 4

1198971015840

119898

sum119895=2

(4119899([4(119898minus1

sum119894=1

1198971015840119894minus 119898 + 1) + 4119895 minus 1]

times(41198971015840119898

minus 4119895 + 3))

minus1

)

12

+119898minus1

sum119896=2

(1198971015840119896+ 1)

times (4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1])

minus1

)

12

(12)

Proof The cuts in 1198676119899are divided into two types I-type and

II-type (see Figure 4) An edge is called I-type if it intersectswith I-type cut Also an edge is called II-type if it intersectswith II-type cut In what follows we consider two situationsaccording to the type of an edge

Case 1 If edge 119890 is I-type in 119895th square of 119896th fragment (ie 119890is edge which is passed by dotted line in Figure 4)

Subcase 11 If 119896 = 1 then we have

1198991(119890) = 21198971015840

1+ 1

1198992(119890) = 4

119898

sum119894=2

(1198971015840119894minus 119898 + 1) + 21198971015840

1+ 1

(13)

Subcase 12 If 119896 = 119898 then we obtain

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119898 + 1) + 21198971015840

119896+ 1

1198992(119890) = 21198971015840

119898+ 1

(14)

Subcase 13 If 2 le 119896 le 119898 minus 1 then we get

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1

1198992(119890) = 4

119898

sum119894=119896+1

(1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1

(15)

Case 2 If edge 119890 is II-type in 119895th square of 119896th fragment (ie119890 is the edge which is passed by real line in Figure 4)

Subcase 21 If 119896 = 1 then we yield

1198991(119890) = 4119895 minus 1

1198992(119890) = 4

119898

sum119894=2

1198971015840119894+ 411989710158401minus 4119895 + 3

(16)

Subcase 22 If 119896 = 119898 then we infer

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 41198971015840

119898minus 4119895 + 3

(17)

Subcase 23 If 2 le 119896 le 119898 minus 1 then we deduce

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 4

119898

sum119894=119896+1

1198971015840119894+ 4 (1198971015840119896minus 119895) + 3

(18)

In particular if 119895 = 1198971015840119896in Subcase 23 we have 119899

1(119890) =

4sum119896

119894=1(1198971015840119894minus 119896 + 1) minus 1 and 119899

2(119890) = 4sum

119898

119894=119896+1(1198971015840119894minus 119898 + 119896) + 3

Hence we get the desired result by combining the abovecases and the definition of the second atom-bond connectiv-ity index

Corollary 5 Let 1198716119899be the linear chain with 119899 hexagonal

Then one has

1198601198611198622(1198716119899) = 4119899minus1

sum119895=1

radic4119899

(4119895 minus 1) (4119899 minus 4119895 + 3) (19)

Proof By the definition of linear chain we check that 119898 =1 11989710158401= 119899 11989710158402= sdot sdot sdot = 1198971015840

119898= 0 In terms ofTheorem 4 we shortly

get the result

Corollary 6 Let 1198856119899be the zig-zag chain with 119899 hexagonal

Then one has

1198601198611198622(1198856119899) = 8radic

4119899

3 (4119899 minus 1)

+ 2119899minus2

sum119896=1

radic4119899

(4119896 + 3) (4119899 + 4119896 minus 1)

+ 3119899minus2

sum119896=2

radic4119899

(4119896 + 1) (4119899 minus 4119896 + 1)

(20)

Proof Using the definition of zig-zag chain we verify that119898 = 119899 minus 1 and 1198971015840

1= 11989710158402= sdot sdot sdot = 1198971015840

119898= 2 In view of Theorem 4

the corollary is immediately yielded

4 Second Atom-Bond Connectivity Index ofV-Phenylenic Nanotubes and Nanotori

The notations in this section follow from Diudea [15]In what follows we use 119899(119906) and 119899(V) to express 119899

1(119890)

and 1198992(119890) for short respectively The molecular structures

V-phenylenic nanotube and V-phenylenic nanotorus aredenoted by 119881119875119867119883[119898 119899] and 119881119875119867119884[119898 119899] respectively

The next result shows the representation of1198601198611198622(119881119875119867119883[119898 119899]) and the structure of 119881119875119867119883[119898 119899]

is described in Figure 5

6 Journal of Chemistry

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 5 The structure of V-phenylenic nanotube

Theorem 7 Consider

1198601198611198622(119881119875119867119883 [119898 119899]) =

1205721 if 119898 = 119899

1205722 if 119898 = 119899

(21)

where 1205721and 120572

2are presented in the process of the proof

Proof First we note that |119881(119881119875119867119883[119898 119899])| = 6119898119899To compute the second atom-bond connectivity index of119881119875119867119883[119898 119899] we assume that 119864

1 1198642 and 119864

3are the set of all

vertical oblique and horizontal edges respectivelyThen weinfer

1198601198611198622(119881119875119867119883 [119898 119899])

= sum119890isin119864(119881119875119867119883[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(22)

Now we calculate sum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) by con-sidering the following two cases

Case 1 (119898 = 119899) Under this situation we deduce

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(23)where 119878

119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

in this case we infer that the second atom-bond connectivityindex of 119881119875119867119883[119898 119899] is

1205721= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(24)Case 2 (119898 = 119899) In this case we get

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(25)

Thus the corresponding second atom-bond connectivityindex is

1205722= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(26)The result is archived

Journal of Chemistry 7

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 6 The structure of V-phenylenic nanotorus

Our last result computes the second atom-bond connec-tivity index of 119881119875119867119884[119898 119899] and the structure of 119881119875119867119884[119898 119899]presented in Figure 6

Theorem 8 Consider

1198601198611198622(119881119875119867119884 [119898 119899]) =

1205723 if 119898 = 119899

1205724 if 119898 = 119899

(27)

where 1205723and 120572

4are presented in the process of the proof

Proof In order to prove the theorem we use a similar trickas in Theorem 7 Obviously |119881(119881119875119867119884[119898 119899])| = 6119898119899 Weassume that 119864

1 1198642 and 119864

3are the set of all vertical oblique

and horizontal edges respectively Then we deduce

1198601198611198622(119881119875119867119884 [119898 119899])

= sum119890isin119864(119881119875119867119884[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(28)

Now we calculatesum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) accord-ing to the relationship between 119898 and 119899

Case 1 (119898 = 119899) Under this assumption we infer

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ( (2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3)119894))

minus1

)

12

(29)

where 119878119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

the second atom-bond connectivity index of119881119875119867119884[119898 119899] for119898 = 119899 is

1205723= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894))

minus1

)12

(30)

Case 2 (119898 = 119899) In this case we have

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(31)

8 Journal of Chemistry

Thus the corresponding second atom-bond connectivityindex is

1205724= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(32)

The proof is completed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for their constructive com-ments in improving the quality of this paperThey also wouldlike to thank the anonymous referees for providing themwith constructive comments and suggestions The researchis partially supported by NSFC (nos 11401519 11371328 and11471293)

References

[1] L Yan Y Li W Gao and J Li ldquoPI index for some specialgraphsrdquo Journal of Chemical and Pharmaceutical Research vol5 no 11 pp 260ndash264 2013

[2] L Yan Y Li W Gao and J Li ldquoOn the extremal hyper-wienerindex of graphsrdquo Journal of Chemical and PharmaceuticalResearch vol 6 no 3 pp 477ndash481 2014

[3] W Gao and L Shi ldquoWiener index of gear fan graph and gearwheel graphrdquo Asian Journal of Chemistry vol 26 no 11 pp3397ndash3400 2014

[4] W F Xi and W Gao ldquoGeometric-arithmetic index and Zagrebindices of certain special molecular graphsrdquo Journal of Advancesin Chemistry vol 10 no 2 pp 2254ndash2261 2014

[5] J Y Dou Y Y Wang and W Gao ldquoSome characteristicson hyper-wiener index of graphsrdquo Journal of Chemical andPharmaceutical Research vol 6 no 5 pp 1659ndash1663 2014

[6] J A Bondy and U S R Murty GraphTheory Springer Berlin Germany 2008

[7] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry A vol 37 no 10 pp849ndash855 1998

[8] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquoChemical Physics Letters vol 511 pp 452ndash454 2011

[9] B Furtula A Graovac and D Vukicevic ldquoAtom-bond connec-tivity index of treesrdquo Discrete Applied Mathematics vol 157 no13 pp 2828ndash2835 2009

[10] T S Vassilev and L J Huntington ldquoOn the minimum ABCindex of chemical treesrdquo Applied Mathematics vol 2 no 1 pp8ndash16 2012

[11] J Chen J Liu and Q Li ldquoThe atom-bond connectivity index ofcatacondensed polyomino graphsrdquoDiscrete Dynamics inNatureand Society vol 2013 Article ID 598517 7 pages 2013

[12] A Graovac and M Ghorbani ldquoA new version of atom-bondconnectivity indexrdquo Acta Chimica Slovenica vol 57 no 3 pp609ndash612 2010

[13] M Rostami M S Haghighat and M Ghorbani ldquoOn secondatom-bond connectivity indexrdquo Iranian Journal of Mathemati-cal Chemistry vol 4 no 2 pp 265ndash270 2013

[14] I Gutman and S Klavzar ldquoAn algorithm for the calculationof the Szeged index of benzenoid hydrocarbonsrdquo Journal ofChemical Information and Computer Sciences vol 35 no 6 pp1011ndash1014 1995

[15] M V Diudea ldquoPhenylenic and naphthylenic torirdquo FullerenesNanotubes and Carbon Nanostructures vol 10 no 4 pp 273ndash292 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal ofPhotoenergy

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Carbohydrate Chemistry

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Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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Hindawi Publishing Corporationhttpwwwhindawicom

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Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

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Chromatography Research International

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Theoretical ChemistryJournal of

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Journal of

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Quantum Chemistry

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Organic Chemistry International

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CatalystsJournal of

Journal of Chemistry 3

II III

I CP1 P2

II II I

I

I

II II

middot middot middot

middot middot middot

middot middot middot

Figure 2 I-type cut and II-type cut of unilateral polyomino chain

+119898

sum119896=1

(119897119896+ 1)

times (2119899((2119896minus1

sum119894=1

119897119894minus 2119896 + 119897

119896+ 3)

times (2119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 119897

119896+ 1))

minus1

)

12

(3)

Proof The cuts in 1198674119899are divided into two types I-type and

II-type (see Figure 2) An edge is called I-type (or II-type) ifit intersects with I-type (or II-type) cut Now we consider thefollowing two cases

Case 1 If edge 119890 is I-type in 119895th square of 119896th fragment (ie 119890is edge which is passed by dotted line in Figure 2) we observethat there is 119897

119896+ 1 such edges in 119896th fragment

Subcase 11 If 119896 = 1 then we have

1198991(119890) = 119897

1+ 1

1198992(119890) = 2

119898

sum119894=2

119897119894minus 2119898 + 119897

1+ 3

(4)

Subcase 12 If 119896 = 119898 then we obtain

1198991(119890) = 2

119898minus1

sum119894=1

119897119894minus 2119898 + 119897

119898+ 3

1198992(119890) = 119897

119898+ 1

(5)

Subcase 13 If 2 le 119896 le 119898 minus 1 then we get

1198991(119890) = 2

119896minus1

sum119894=1

119897119894minus 2119896 + 119897

119896+ 3

1198992(119890) = 2

119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 119897

119896+ 1

(6)

Case 2 If edge 119890 is II-type in 119895th square of 119896th fragment (ie119890 is the edge which is passed by real line in Figure 2)

Subcase 21 If 119896 = 1 then we yield

1198991(119890) = 2119895

1198992(119890) = 2

119898

sum119894=2

119897119894minus 2119898 + 2 (119897

1minus 119895) + 4

(7)

Subcase 22 If 119896 = 119898 then we infer

1198991(119890) = 2

119898minus1

sum119894=1

119897119894minus 2119898 + 2119895 + 2

1198992(119890) = 2119897

119898minus 2119895 + 2

(8)

Subcase 23 If 2 le 119896 le 119898 minus 1 then we deduce

1198991(119890) = 2

119896minus1

sum119894=1

119897119894minus 2119896 + 2119895 + 2

1198992(119890) = 2

119898

sum119894=119896+1

119897119894minus 2 (119898 minus 119896) + 2 (119897

119896minus 119895) + 2

(9)

Hence the desired conclusion is obtained by combiningthe above case and the definition of the second atom-bondconnectivity index

Corollary 2 Let 1198714119899be the linear chain with 119899 squares Then

one has

1198601198611198622(1198714119899) = 2119899minus1

sum119895=1

radic119899

2119895 (119899 minus 119895 + 1)+ radic2119899 (10)

Proof By the definition of linear chain we have 119898 = 1 1198971=

119899 1198972= sdot sdot sdot = 119897

119898= 0 In terms of Theorem 1 we immediately

get the result

Corollary 3 Let1198854119899be the zig-zag chain with 119899 squaresThen

one has

1198601198611198622(1198854119899) = 4radic

1

2+119898

sum119896=1

3radic2119899

(2119896 + 1) (2119899 minus 2119896 + 1) (11)

Proof By the virtue of the definition of the zig-zag chain wehave 119898 = 119899 minus 1 and 119897

1= 1198972

= sdot sdot sdot = 119897119898

= 2 In view ofTheorem 1 the result is immediately obtained

4 Journal of Chemistry

Linear hexagonal chain Zig-zag hexagonal chain

middot middot middot

middot middot middot

middot middot middot

Figure 3 The structure of 1198716119899and 1198856

119899

3 Second Atom-Bond Connectivity Index ofHexagonal Chain

Hexagonal chain is one class of hexagonal system consistingof hexagonal In hexagonal chain each of two hexagonal hasone common edge or no common vertex Two hexagonalare adjacent if they have common edge No three or morehexagonal share one vertex Each hexagonal has two adjacenthexagonal except hexagonal in terminus and each hexagonalchain has two hexagonal in terminus

It is easy to verify that the hexagonal chain with119899 hexagonal has 4119899 + 2 vertices and 5119899 + 1 edges Let 1198716

119899and

1198856119899be the linear hexagonal chain and zig-zag hexagonal chain

respectively For the chemical structure of 1198716119899and1198856

119899 one can

refer to Figure 3 for more detailAgain we use the similar trick which was presented in

Gutman and Klavzar [14] and we define elementary cut asfollows Choose an edge 119890 of the hexagonal system and drawa straight line through the center of 119890 orthogonal on 119890 Thisline will intersect the perimeter in two end points 119875

1and 119875

2

The straight line segment 119862 whose end points are 1198751and 119875

2

is the elementary cut intersecting the edge 119890 A fragment 119878 inhexagonal chain is just maximal linear chain which includesthe hexagonal in start and end vertices Let 1198971015840(119878) be thelength of fragment which denotes the number of hexagonalcontained Let 1198676

119899be a hexagonal chain with 119899 hexagonal

consisting of fragment sequence 1198781 1198782 119878

119898(119898 ge 1)

Denote 1198971015840(119878119894) = 1198971015840119894(119894 = 1 119898) Then we verify that 1198971015840

1+ 11989710158402+

sdot sdot sdot + 1198971015840119898

= 119899 + 119898 minus 1 since each two adjacent fragments haveone common hexagonal For the 119896th fragment of hexagonalchain the cut of this fragment is the cut which intersectswith 1198971015840

119896+ 1 parallel edges of hexagonal in this fragment A

fragment is called horizontal fragment if its cut parallels thehorizontal direction otherwise it is called inclined fragmentUnilateral hexagonal chain is a special class of hexagonalchain such that the cut for each inclined fragment at the sameangle with a horizontal direction As an example Figure 4shows a structure of unilateral hexagonal chain Clearlylinear hexagonal chain 1198716

119899is a unilateral hexagonal chain with

one fragment and zig-zag is a unilateral hexagonal chainwith119899 minus 1 fragments

I II

IC

II

II

IIII

II

II

II

II

II I

I

I

P1 P2

middot middot middot

middot middot middot

⋱

Figure 4 I-type cut and II-type cut of unilateral hexagonal chain

We now show the second atom-bond connectivity indexof unilateral hexagonal chain

Theorem 4 Let1198676119899be a unilateral hexagonal chain consisting

of 119898 fragment 1198781 1198782 119878

119898(119898 ge 1) and let 1198971015840(119878

119894) = 1198971015840119894(119894 =

1 119898) be the length of each fragment Then one has

1198601198611198622(1198676119899)

= 4

1198971015840

1minus1

sum119895=1

radic4119899

(4119895 minus 1) [4 (sum119898

119894=21198971015840119894minus 119898 + 1) + 41198971015840

1minus 4119895 + 3]

+ 2119898minus1

sum119896=1

(4119899([4(119896

sum119894=1

1198971015840119894minus 119896 + 1) minus 1]

times[4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 3])

minus1

)

12

+ 4119898minus1

sum119896=2

1198971015840

119896minus1

sum119895=2

(4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 4119895 minus 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896)

+41198971015840119896minus 4119895 + 3])

minus1

)

12

Journal of Chemistry 5

+ 4

1198971015840

119898

sum119895=2

(4119899([4(119898minus1

sum119894=1

1198971015840119894minus 119898 + 1) + 4119895 minus 1]

times(41198971015840119898

minus 4119895 + 3))

minus1

)

12

+119898minus1

sum119896=2

(1198971015840119896+ 1)

times (4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1])

minus1

)

12

(12)

Proof The cuts in 1198676119899are divided into two types I-type and

II-type (see Figure 4) An edge is called I-type if it intersectswith I-type cut Also an edge is called II-type if it intersectswith II-type cut In what follows we consider two situationsaccording to the type of an edge

Case 1 If edge 119890 is I-type in 119895th square of 119896th fragment (ie 119890is edge which is passed by dotted line in Figure 4)

Subcase 11 If 119896 = 1 then we have

1198991(119890) = 21198971015840

1+ 1

1198992(119890) = 4

119898

sum119894=2

(1198971015840119894minus 119898 + 1) + 21198971015840

1+ 1

(13)

Subcase 12 If 119896 = 119898 then we obtain

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119898 + 1) + 21198971015840

119896+ 1

1198992(119890) = 21198971015840

119898+ 1

(14)

Subcase 13 If 2 le 119896 le 119898 minus 1 then we get

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1

1198992(119890) = 4

119898

sum119894=119896+1

(1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1

(15)

Case 2 If edge 119890 is II-type in 119895th square of 119896th fragment (ie119890 is the edge which is passed by real line in Figure 4)

Subcase 21 If 119896 = 1 then we yield

1198991(119890) = 4119895 minus 1

1198992(119890) = 4

119898

sum119894=2

1198971015840119894+ 411989710158401minus 4119895 + 3

(16)

Subcase 22 If 119896 = 119898 then we infer

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 41198971015840

119898minus 4119895 + 3

(17)

Subcase 23 If 2 le 119896 le 119898 minus 1 then we deduce

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 4

119898

sum119894=119896+1

1198971015840119894+ 4 (1198971015840119896minus 119895) + 3

(18)

In particular if 119895 = 1198971015840119896in Subcase 23 we have 119899

1(119890) =

4sum119896

119894=1(1198971015840119894minus 119896 + 1) minus 1 and 119899

2(119890) = 4sum

119898

119894=119896+1(1198971015840119894minus 119898 + 119896) + 3

Hence we get the desired result by combining the abovecases and the definition of the second atom-bond connectiv-ity index

Corollary 5 Let 1198716119899be the linear chain with 119899 hexagonal

Then one has

1198601198611198622(1198716119899) = 4119899minus1

sum119895=1

radic4119899

(4119895 minus 1) (4119899 minus 4119895 + 3) (19)

Proof By the definition of linear chain we check that 119898 =1 11989710158401= 119899 11989710158402= sdot sdot sdot = 1198971015840

119898= 0 In terms ofTheorem 4 we shortly

get the result

Corollary 6 Let 1198856119899be the zig-zag chain with 119899 hexagonal

Then one has

1198601198611198622(1198856119899) = 8radic

4119899

3 (4119899 minus 1)

+ 2119899minus2

sum119896=1

radic4119899

(4119896 + 3) (4119899 + 4119896 minus 1)

+ 3119899minus2

sum119896=2

radic4119899

(4119896 + 1) (4119899 minus 4119896 + 1)

(20)

Proof Using the definition of zig-zag chain we verify that119898 = 119899 minus 1 and 1198971015840

1= 11989710158402= sdot sdot sdot = 1198971015840

119898= 2 In view of Theorem 4

the corollary is immediately yielded

4 Second Atom-Bond Connectivity Index ofV-Phenylenic Nanotubes and Nanotori

The notations in this section follow from Diudea [15]In what follows we use 119899(119906) and 119899(V) to express 119899

1(119890)

and 1198992(119890) for short respectively The molecular structures

V-phenylenic nanotube and V-phenylenic nanotorus aredenoted by 119881119875119867119883[119898 119899] and 119881119875119867119884[119898 119899] respectively

The next result shows the representation of1198601198611198622(119881119875119867119883[119898 119899]) and the structure of 119881119875119867119883[119898 119899]

is described in Figure 5

6 Journal of Chemistry

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 5 The structure of V-phenylenic nanotube

Theorem 7 Consider

1198601198611198622(119881119875119867119883 [119898 119899]) =

1205721 if 119898 = 119899

1205722 if 119898 = 119899

(21)

where 1205721and 120572

2are presented in the process of the proof

Proof First we note that |119881(119881119875119867119883[119898 119899])| = 6119898119899To compute the second atom-bond connectivity index of119881119875119867119883[119898 119899] we assume that 119864

1 1198642 and 119864

3are the set of all

vertical oblique and horizontal edges respectivelyThen weinfer

1198601198611198622(119881119875119867119883 [119898 119899])

= sum119890isin119864(119881119875119867119883[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(22)

Now we calculate sum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) by con-sidering the following two cases

Case 1 (119898 = 119899) Under this situation we deduce

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(23)where 119878

119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

in this case we infer that the second atom-bond connectivityindex of 119881119875119867119883[119898 119899] is

1205721= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(24)Case 2 (119898 = 119899) In this case we get

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(25)

Thus the corresponding second atom-bond connectivityindex is

1205722= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(26)The result is archived

Journal of Chemistry 7

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 6 The structure of V-phenylenic nanotorus

Our last result computes the second atom-bond connec-tivity index of 119881119875119867119884[119898 119899] and the structure of 119881119875119867119884[119898 119899]presented in Figure 6

Theorem 8 Consider

1198601198611198622(119881119875119867119884 [119898 119899]) =

1205723 if 119898 = 119899

1205724 if 119898 = 119899

(27)

where 1205723and 120572

4are presented in the process of the proof

Proof In order to prove the theorem we use a similar trickas in Theorem 7 Obviously |119881(119881119875119867119884[119898 119899])| = 6119898119899 Weassume that 119864

1 1198642 and 119864

3are the set of all vertical oblique

and horizontal edges respectively Then we deduce

1198601198611198622(119881119875119867119884 [119898 119899])

= sum119890isin119864(119881119875119867119884[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(28)

Now we calculatesum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) accord-ing to the relationship between 119898 and 119899

Case 1 (119898 = 119899) Under this assumption we infer

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ( (2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3)119894))

minus1

)

12

(29)

where 119878119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

the second atom-bond connectivity index of119881119875119867119884[119898 119899] for119898 = 119899 is

1205723= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894))

minus1

)12

(30)

Case 2 (119898 = 119899) In this case we have

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(31)

8 Journal of Chemistry

Thus the corresponding second atom-bond connectivityindex is

1205724= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(32)

The proof is completed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for their constructive com-ments in improving the quality of this paperThey also wouldlike to thank the anonymous referees for providing themwith constructive comments and suggestions The researchis partially supported by NSFC (nos 11401519 11371328 and11471293)

References

[1] L Yan Y Li W Gao and J Li ldquoPI index for some specialgraphsrdquo Journal of Chemical and Pharmaceutical Research vol5 no 11 pp 260ndash264 2013

[2] L Yan Y Li W Gao and J Li ldquoOn the extremal hyper-wienerindex of graphsrdquo Journal of Chemical and PharmaceuticalResearch vol 6 no 3 pp 477ndash481 2014

[3] W Gao and L Shi ldquoWiener index of gear fan graph and gearwheel graphrdquo Asian Journal of Chemistry vol 26 no 11 pp3397ndash3400 2014

[4] W F Xi and W Gao ldquoGeometric-arithmetic index and Zagrebindices of certain special molecular graphsrdquo Journal of Advancesin Chemistry vol 10 no 2 pp 2254ndash2261 2014

[5] J Y Dou Y Y Wang and W Gao ldquoSome characteristicson hyper-wiener index of graphsrdquo Journal of Chemical andPharmaceutical Research vol 6 no 5 pp 1659ndash1663 2014

[6] J A Bondy and U S R Murty GraphTheory Springer Berlin Germany 2008

[7] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry A vol 37 no 10 pp849ndash855 1998

[8] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquoChemical Physics Letters vol 511 pp 452ndash454 2011

[9] B Furtula A Graovac and D Vukicevic ldquoAtom-bond connec-tivity index of treesrdquo Discrete Applied Mathematics vol 157 no13 pp 2828ndash2835 2009

[10] T S Vassilev and L J Huntington ldquoOn the minimum ABCindex of chemical treesrdquo Applied Mathematics vol 2 no 1 pp8ndash16 2012

[11] J Chen J Liu and Q Li ldquoThe atom-bond connectivity index ofcatacondensed polyomino graphsrdquoDiscrete Dynamics inNatureand Society vol 2013 Article ID 598517 7 pages 2013

[12] A Graovac and M Ghorbani ldquoA new version of atom-bondconnectivity indexrdquo Acta Chimica Slovenica vol 57 no 3 pp609ndash612 2010

[13] M Rostami M S Haghighat and M Ghorbani ldquoOn secondatom-bond connectivity indexrdquo Iranian Journal of Mathemati-cal Chemistry vol 4 no 2 pp 265ndash270 2013

[14] I Gutman and S Klavzar ldquoAn algorithm for the calculationof the Szeged index of benzenoid hydrocarbonsrdquo Journal ofChemical Information and Computer Sciences vol 35 no 6 pp1011ndash1014 1995

[15] M V Diudea ldquoPhenylenic and naphthylenic torirdquo FullerenesNanotubes and Carbon Nanostructures vol 10 no 4 pp 273ndash292 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

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Medicinal ChemistryInternational Journal of

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Chromatography Research International

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CatalystsJournal of

4 Journal of Chemistry

Linear hexagonal chain Zig-zag hexagonal chain

middot middot middot

middot middot middot

middot middot middot

Figure 3 The structure of 1198716119899and 1198856

119899

3 Second Atom-Bond Connectivity Index ofHexagonal Chain

Hexagonal chain is one class of hexagonal system consistingof hexagonal In hexagonal chain each of two hexagonal hasone common edge or no common vertex Two hexagonalare adjacent if they have common edge No three or morehexagonal share one vertex Each hexagonal has two adjacenthexagonal except hexagonal in terminus and each hexagonalchain has two hexagonal in terminus

It is easy to verify that the hexagonal chain with119899 hexagonal has 4119899 + 2 vertices and 5119899 + 1 edges Let 1198716

119899and

1198856119899be the linear hexagonal chain and zig-zag hexagonal chain

respectively For the chemical structure of 1198716119899and1198856

119899 one can

refer to Figure 3 for more detailAgain we use the similar trick which was presented in

Gutman and Klavzar [14] and we define elementary cut asfollows Choose an edge 119890 of the hexagonal system and drawa straight line through the center of 119890 orthogonal on 119890 Thisline will intersect the perimeter in two end points 119875

1and 119875

2

The straight line segment 119862 whose end points are 1198751and 119875

2

is the elementary cut intersecting the edge 119890 A fragment 119878 inhexagonal chain is just maximal linear chain which includesthe hexagonal in start and end vertices Let 1198971015840(119878) be thelength of fragment which denotes the number of hexagonalcontained Let 1198676

119899be a hexagonal chain with 119899 hexagonal

consisting of fragment sequence 1198781 1198782 119878

119898(119898 ge 1)

Denote 1198971015840(119878119894) = 1198971015840119894(119894 = 1 119898) Then we verify that 1198971015840

1+ 11989710158402+

sdot sdot sdot + 1198971015840119898

= 119899 + 119898 minus 1 since each two adjacent fragments haveone common hexagonal For the 119896th fragment of hexagonalchain the cut of this fragment is the cut which intersectswith 1198971015840

119896+ 1 parallel edges of hexagonal in this fragment A

fragment is called horizontal fragment if its cut parallels thehorizontal direction otherwise it is called inclined fragmentUnilateral hexagonal chain is a special class of hexagonalchain such that the cut for each inclined fragment at the sameangle with a horizontal direction As an example Figure 4shows a structure of unilateral hexagonal chain Clearlylinear hexagonal chain 1198716

119899is a unilateral hexagonal chain with

one fragment and zig-zag is a unilateral hexagonal chainwith119899 minus 1 fragments

I II

IC

II

II

IIII

II

II

II

II

II I

I

I

P1 P2

middot middot middot

middot middot middot

⋱

Figure 4 I-type cut and II-type cut of unilateral hexagonal chain

We now show the second atom-bond connectivity indexof unilateral hexagonal chain

Theorem 4 Let1198676119899be a unilateral hexagonal chain consisting

of 119898 fragment 1198781 1198782 119878

119898(119898 ge 1) and let 1198971015840(119878

119894) = 1198971015840119894(119894 =

1 119898) be the length of each fragment Then one has

1198601198611198622(1198676119899)

= 4

1198971015840

1minus1

sum119895=1

radic4119899

(4119895 minus 1) [4 (sum119898

119894=21198971015840119894minus 119898 + 1) + 41198971015840

1minus 4119895 + 3]

+ 2119898minus1

sum119896=1

(4119899([4(119896

sum119894=1

1198971015840119894minus 119896 + 1) minus 1]

times[4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 3])

minus1

)

12

+ 4119898minus1

sum119896=2

1198971015840

119896minus1

sum119895=2

(4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 4119895 minus 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896)

+41198971015840119896minus 4119895 + 3])

minus1

)

12

Journal of Chemistry 5

+ 4

1198971015840

119898

sum119895=2

(4119899([4(119898minus1

sum119894=1

1198971015840119894minus 119898 + 1) + 4119895 minus 1]

times(41198971015840119898

minus 4119895 + 3))

minus1

)

12

+119898minus1

sum119896=2

(1198971015840119896+ 1)

times (4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1])

minus1

)

12

(12)

Proof The cuts in 1198676119899are divided into two types I-type and

II-type (see Figure 4) An edge is called I-type if it intersectswith I-type cut Also an edge is called II-type if it intersectswith II-type cut In what follows we consider two situationsaccording to the type of an edge

Case 1 If edge 119890 is I-type in 119895th square of 119896th fragment (ie 119890is edge which is passed by dotted line in Figure 4)

Subcase 11 If 119896 = 1 then we have

1198991(119890) = 21198971015840

1+ 1

1198992(119890) = 4

119898

sum119894=2

(1198971015840119894minus 119898 + 1) + 21198971015840

1+ 1

(13)

Subcase 12 If 119896 = 119898 then we obtain

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119898 + 1) + 21198971015840

119896+ 1

1198992(119890) = 21198971015840

119898+ 1

(14)

Subcase 13 If 2 le 119896 le 119898 minus 1 then we get

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1

1198992(119890) = 4

119898

sum119894=119896+1

(1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1

(15)

Case 2 If edge 119890 is II-type in 119895th square of 119896th fragment (ie119890 is the edge which is passed by real line in Figure 4)

Subcase 21 If 119896 = 1 then we yield

1198991(119890) = 4119895 minus 1

1198992(119890) = 4

119898

sum119894=2

1198971015840119894+ 411989710158401minus 4119895 + 3

(16)

Subcase 22 If 119896 = 119898 then we infer

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 41198971015840

119898minus 4119895 + 3

(17)

Subcase 23 If 2 le 119896 le 119898 minus 1 then we deduce

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 4

119898

sum119894=119896+1

1198971015840119894+ 4 (1198971015840119896minus 119895) + 3

(18)

In particular if 119895 = 1198971015840119896in Subcase 23 we have 119899

1(119890) =

4sum119896

119894=1(1198971015840119894minus 119896 + 1) minus 1 and 119899

2(119890) = 4sum

119898

119894=119896+1(1198971015840119894minus 119898 + 119896) + 3

Hence we get the desired result by combining the abovecases and the definition of the second atom-bond connectiv-ity index

Corollary 5 Let 1198716119899be the linear chain with 119899 hexagonal

Then one has

1198601198611198622(1198716119899) = 4119899minus1

sum119895=1

radic4119899

(4119895 minus 1) (4119899 minus 4119895 + 3) (19)

Proof By the definition of linear chain we check that 119898 =1 11989710158401= 119899 11989710158402= sdot sdot sdot = 1198971015840

119898= 0 In terms ofTheorem 4 we shortly

get the result

Corollary 6 Let 1198856119899be the zig-zag chain with 119899 hexagonal

Then one has

1198601198611198622(1198856119899) = 8radic

4119899

3 (4119899 minus 1)

+ 2119899minus2

sum119896=1

radic4119899

(4119896 + 3) (4119899 + 4119896 minus 1)

+ 3119899minus2

sum119896=2

radic4119899

(4119896 + 1) (4119899 minus 4119896 + 1)

(20)

Proof Using the definition of zig-zag chain we verify that119898 = 119899 minus 1 and 1198971015840

1= 11989710158402= sdot sdot sdot = 1198971015840

119898= 2 In view of Theorem 4

the corollary is immediately yielded

4 Second Atom-Bond Connectivity Index ofV-Phenylenic Nanotubes and Nanotori

The notations in this section follow from Diudea [15]In what follows we use 119899(119906) and 119899(V) to express 119899

1(119890)

and 1198992(119890) for short respectively The molecular structures

V-phenylenic nanotube and V-phenylenic nanotorus aredenoted by 119881119875119867119883[119898 119899] and 119881119875119867119884[119898 119899] respectively

The next result shows the representation of1198601198611198622(119881119875119867119883[119898 119899]) and the structure of 119881119875119867119883[119898 119899]

is described in Figure 5

6 Journal of Chemistry

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 5 The structure of V-phenylenic nanotube

Theorem 7 Consider

1198601198611198622(119881119875119867119883 [119898 119899]) =

1205721 if 119898 = 119899

1205722 if 119898 = 119899

(21)

where 1205721and 120572

2are presented in the process of the proof

Proof First we note that |119881(119881119875119867119883[119898 119899])| = 6119898119899To compute the second atom-bond connectivity index of119881119875119867119883[119898 119899] we assume that 119864

1 1198642 and 119864

3are the set of all

vertical oblique and horizontal edges respectivelyThen weinfer

1198601198611198622(119881119875119867119883 [119898 119899])

= sum119890isin119864(119881119875119867119883[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(22)

Now we calculate sum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) by con-sidering the following two cases

Case 1 (119898 = 119899) Under this situation we deduce

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(23)where 119878

119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

in this case we infer that the second atom-bond connectivityindex of 119881119875119867119883[119898 119899] is

1205721= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(24)Case 2 (119898 = 119899) In this case we get

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(25)

Thus the corresponding second atom-bond connectivityindex is

1205722= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(26)The result is archived

Journal of Chemistry 7

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 6 The structure of V-phenylenic nanotorus

Our last result computes the second atom-bond connec-tivity index of 119881119875119867119884[119898 119899] and the structure of 119881119875119867119884[119898 119899]presented in Figure 6

Theorem 8 Consider

1198601198611198622(119881119875119867119884 [119898 119899]) =

1205723 if 119898 = 119899

1205724 if 119898 = 119899

(27)

where 1205723and 120572

4are presented in the process of the proof

Proof In order to prove the theorem we use a similar trickas in Theorem 7 Obviously |119881(119881119875119867119884[119898 119899])| = 6119898119899 Weassume that 119864

1 1198642 and 119864

3are the set of all vertical oblique

and horizontal edges respectively Then we deduce

1198601198611198622(119881119875119867119884 [119898 119899])

= sum119890isin119864(119881119875119867119884[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(28)

Now we calculatesum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) accord-ing to the relationship between 119898 and 119899

Case 1 (119898 = 119899) Under this assumption we infer

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ( (2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3)119894))

minus1

)

12

(29)

where 119878119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

the second atom-bond connectivity index of119881119875119867119884[119898 119899] for119898 = 119899 is

1205723= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894))

minus1

)12

(30)

Case 2 (119898 = 119899) In this case we have

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(31)

8 Journal of Chemistry

Thus the corresponding second atom-bond connectivityindex is

1205724= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(32)

The proof is completed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for their constructive com-ments in improving the quality of this paperThey also wouldlike to thank the anonymous referees for providing themwith constructive comments and suggestions The researchis partially supported by NSFC (nos 11401519 11371328 and11471293)

References

[1] L Yan Y Li W Gao and J Li ldquoPI index for some specialgraphsrdquo Journal of Chemical and Pharmaceutical Research vol5 no 11 pp 260ndash264 2013

[2] L Yan Y Li W Gao and J Li ldquoOn the extremal hyper-wienerindex of graphsrdquo Journal of Chemical and PharmaceuticalResearch vol 6 no 3 pp 477ndash481 2014

[3] W Gao and L Shi ldquoWiener index of gear fan graph and gearwheel graphrdquo Asian Journal of Chemistry vol 26 no 11 pp3397ndash3400 2014

[4] W F Xi and W Gao ldquoGeometric-arithmetic index and Zagrebindices of certain special molecular graphsrdquo Journal of Advancesin Chemistry vol 10 no 2 pp 2254ndash2261 2014

[5] J Y Dou Y Y Wang and W Gao ldquoSome characteristicson hyper-wiener index of graphsrdquo Journal of Chemical andPharmaceutical Research vol 6 no 5 pp 1659ndash1663 2014

[6] J A Bondy and U S R Murty GraphTheory Springer Berlin Germany 2008

[7] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry A vol 37 no 10 pp849ndash855 1998

[8] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquoChemical Physics Letters vol 511 pp 452ndash454 2011

[9] B Furtula A Graovac and D Vukicevic ldquoAtom-bond connec-tivity index of treesrdquo Discrete Applied Mathematics vol 157 no13 pp 2828ndash2835 2009

[10] T S Vassilev and L J Huntington ldquoOn the minimum ABCindex of chemical treesrdquo Applied Mathematics vol 2 no 1 pp8ndash16 2012

[11] J Chen J Liu and Q Li ldquoThe atom-bond connectivity index ofcatacondensed polyomino graphsrdquoDiscrete Dynamics inNatureand Society vol 2013 Article ID 598517 7 pages 2013

[12] A Graovac and M Ghorbani ldquoA new version of atom-bondconnectivity indexrdquo Acta Chimica Slovenica vol 57 no 3 pp609ndash612 2010

[13] M Rostami M S Haghighat and M Ghorbani ldquoOn secondatom-bond connectivity indexrdquo Iranian Journal of Mathemati-cal Chemistry vol 4 no 2 pp 265ndash270 2013

[14] I Gutman and S Klavzar ldquoAn algorithm for the calculationof the Szeged index of benzenoid hydrocarbonsrdquo Journal ofChemical Information and Computer Sciences vol 35 no 6 pp1011ndash1014 1995

[15] M V Diudea ldquoPhenylenic and naphthylenic torirdquo FullerenesNanotubes and Carbon Nanostructures vol 10 no 4 pp 273ndash292 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Medicinal ChemistryInternational Journal of

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CatalystsJournal of

Journal of Chemistry 5

+ 4

1198971015840

119898

sum119895=2

(4119899([4(119898minus1

sum119894=1

1198971015840119894minus 119898 + 1) + 4119895 minus 1]

times(41198971015840119898

minus 4119895 + 3))

minus1

)

12

+119898minus1

sum119896=2

(1198971015840119896+ 1)

times (4119899([4(119896minus1

sum119894=1

1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1]

times [4(119898

sum119894=119896+1

1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1])

minus1

)

12

(12)

Proof The cuts in 1198676119899are divided into two types I-type and

II-type (see Figure 4) An edge is called I-type if it intersectswith I-type cut Also an edge is called II-type if it intersectswith II-type cut In what follows we consider two situationsaccording to the type of an edge

Case 1 If edge 119890 is I-type in 119895th square of 119896th fragment (ie 119890is edge which is passed by dotted line in Figure 4)

Subcase 11 If 119896 = 1 then we have

1198991(119890) = 21198971015840

1+ 1

1198992(119890) = 4

119898

sum119894=2

(1198971015840119894minus 119898 + 1) + 21198971015840

1+ 1

(13)

Subcase 12 If 119896 = 119898 then we obtain

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119898 + 1) + 21198971015840

119896+ 1

1198992(119890) = 21198971015840

119898+ 1

(14)

Subcase 13 If 2 le 119896 le 119898 minus 1 then we get

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 21198971015840

119896+ 1

1198992(119890) = 4

119898

sum119894=119896+1

(1198971015840119894minus 119898 + 119896) + 21198971015840

119896+ 1

(15)

Case 2 If edge 119890 is II-type in 119895th square of 119896th fragment (ie119890 is the edge which is passed by real line in Figure 4)

Subcase 21 If 119896 = 1 then we yield

1198991(119890) = 4119895 minus 1

1198992(119890) = 4

119898

sum119894=2

1198971015840119894+ 411989710158401minus 4119895 + 3

(16)

Subcase 22 If 119896 = 119898 then we infer

1198991(119890) = 4

119898minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 41198971015840

119898minus 4119895 + 3

(17)

Subcase 23 If 2 le 119896 le 119898 minus 1 then we deduce

1198991(119890) = 4

119896minus1

sum119894=1

(1198971015840119894minus 119896 + 1) + 4119895 minus 1

1198992(119890) = 4

119898

sum119894=119896+1

1198971015840119894+ 4 (1198971015840119896minus 119895) + 3

(18)

In particular if 119895 = 1198971015840119896in Subcase 23 we have 119899

1(119890) =

4sum119896

119894=1(1198971015840119894minus 119896 + 1) minus 1 and 119899

2(119890) = 4sum

119898

119894=119896+1(1198971015840119894minus 119898 + 119896) + 3

Hence we get the desired result by combining the abovecases and the definition of the second atom-bond connectiv-ity index

Corollary 5 Let 1198716119899be the linear chain with 119899 hexagonal

Then one has

1198601198611198622(1198716119899) = 4119899minus1

sum119895=1

radic4119899

(4119895 minus 1) (4119899 minus 4119895 + 3) (19)

Proof By the definition of linear chain we check that 119898 =1 11989710158401= 119899 11989710158402= sdot sdot sdot = 1198971015840

119898= 0 In terms ofTheorem 4 we shortly

get the result

Corollary 6 Let 1198856119899be the zig-zag chain with 119899 hexagonal

Then one has

1198601198611198622(1198856119899) = 8radic

4119899

3 (4119899 minus 1)

+ 2119899minus2

sum119896=1

radic4119899

(4119896 + 3) (4119899 + 4119896 minus 1)

+ 3119899minus2

sum119896=2

radic4119899

(4119896 + 1) (4119899 minus 4119896 + 1)

(20)

Proof Using the definition of zig-zag chain we verify that119898 = 119899 minus 1 and 1198971015840

1= 11989710158402= sdot sdot sdot = 1198971015840

119898= 2 In view of Theorem 4

the corollary is immediately yielded

4 Second Atom-Bond Connectivity Index ofV-Phenylenic Nanotubes and Nanotori

The notations in this section follow from Diudea [15]In what follows we use 119899(119906) and 119899(V) to express 119899

1(119890)

and 1198992(119890) for short respectively The molecular structures

V-phenylenic nanotube and V-phenylenic nanotorus aredenoted by 119881119875119867119883[119898 119899] and 119881119875119867119884[119898 119899] respectively

The next result shows the representation of1198601198611198622(119881119875119867119883[119898 119899]) and the structure of 119881119875119867119883[119898 119899]

is described in Figure 5

6 Journal of Chemistry

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 5 The structure of V-phenylenic nanotube

Theorem 7 Consider

1198601198611198622(119881119875119867119883 [119898 119899]) =

1205721 if 119898 = 119899

1205722 if 119898 = 119899

(21)

where 1205721and 120572

2are presented in the process of the proof

Proof First we note that |119881(119881119875119867119883[119898 119899])| = 6119898119899To compute the second atom-bond connectivity index of119881119875119867119883[119898 119899] we assume that 119864

1 1198642 and 119864

3are the set of all

vertical oblique and horizontal edges respectivelyThen weinfer

1198601198611198622(119881119875119867119883 [119898 119899])

= sum119890isin119864(119881119875119867119883[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(22)

Now we calculate sum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) by con-sidering the following two cases

Case 1 (119898 = 119899) Under this situation we deduce

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(23)where 119878

119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

in this case we infer that the second atom-bond connectivityindex of 119881119875119867119883[119898 119899] is

1205721= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(24)Case 2 (119898 = 119899) In this case we get

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(25)

Thus the corresponding second atom-bond connectivityindex is

1205722= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(26)The result is archived

Journal of Chemistry 7

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 6 The structure of V-phenylenic nanotorus

Our last result computes the second atom-bond connec-tivity index of 119881119875119867119884[119898 119899] and the structure of 119881119875119867119884[119898 119899]presented in Figure 6

Theorem 8 Consider

1198601198611198622(119881119875119867119884 [119898 119899]) =

1205723 if 119898 = 119899

1205724 if 119898 = 119899

(27)

where 1205723and 120572

4are presented in the process of the proof

Proof In order to prove the theorem we use a similar trickas in Theorem 7 Obviously |119881(119881119875119867119884[119898 119899])| = 6119898119899 Weassume that 119864

1 1198642 and 119864

3are the set of all vertical oblique

and horizontal edges respectively Then we deduce

1198601198611198622(119881119875119867119884 [119898 119899])

= sum119890isin119864(119881119875119867119884[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(28)

Now we calculatesum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) accord-ing to the relationship between 119898 and 119899

Case 1 (119898 = 119899) Under this assumption we infer

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ( (2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3)119894))

minus1

)

12

(29)

where 119878119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

the second atom-bond connectivity index of119881119875119867119884[119898 119899] for119898 = 119899 is

1205723= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894))

minus1

)12

(30)

Case 2 (119898 = 119899) In this case we have

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(31)

8 Journal of Chemistry

Thus the corresponding second atom-bond connectivityindex is

1205724= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(32)

The proof is completed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for their constructive com-ments in improving the quality of this paperThey also wouldlike to thank the anonymous referees for providing themwith constructive comments and suggestions The researchis partially supported by NSFC (nos 11401519 11371328 and11471293)

References

[1] L Yan Y Li W Gao and J Li ldquoPI index for some specialgraphsrdquo Journal of Chemical and Pharmaceutical Research vol5 no 11 pp 260ndash264 2013

[2] L Yan Y Li W Gao and J Li ldquoOn the extremal hyper-wienerindex of graphsrdquo Journal of Chemical and PharmaceuticalResearch vol 6 no 3 pp 477ndash481 2014

[3] W Gao and L Shi ldquoWiener index of gear fan graph and gearwheel graphrdquo Asian Journal of Chemistry vol 26 no 11 pp3397ndash3400 2014

[4] W F Xi and W Gao ldquoGeometric-arithmetic index and Zagrebindices of certain special molecular graphsrdquo Journal of Advancesin Chemistry vol 10 no 2 pp 2254ndash2261 2014

[5] J Y Dou Y Y Wang and W Gao ldquoSome characteristicson hyper-wiener index of graphsrdquo Journal of Chemical andPharmaceutical Research vol 6 no 5 pp 1659ndash1663 2014

[6] J A Bondy and U S R Murty GraphTheory Springer Berlin Germany 2008

[7] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry A vol 37 no 10 pp849ndash855 1998

[8] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquoChemical Physics Letters vol 511 pp 452ndash454 2011

[9] B Furtula A Graovac and D Vukicevic ldquoAtom-bond connec-tivity index of treesrdquo Discrete Applied Mathematics vol 157 no13 pp 2828ndash2835 2009

[10] T S Vassilev and L J Huntington ldquoOn the minimum ABCindex of chemical treesrdquo Applied Mathematics vol 2 no 1 pp8ndash16 2012

[11] J Chen J Liu and Q Li ldquoThe atom-bond connectivity index ofcatacondensed polyomino graphsrdquoDiscrete Dynamics inNatureand Society vol 2013 Article ID 598517 7 pages 2013

[12] A Graovac and M Ghorbani ldquoA new version of atom-bondconnectivity indexrdquo Acta Chimica Slovenica vol 57 no 3 pp609ndash612 2010

[13] M Rostami M S Haghighat and M Ghorbani ldquoOn secondatom-bond connectivity indexrdquo Iranian Journal of Mathemati-cal Chemistry vol 4 no 2 pp 265ndash270 2013

[14] I Gutman and S Klavzar ldquoAn algorithm for the calculationof the Szeged index of benzenoid hydrocarbonsrdquo Journal ofChemical Information and Computer Sciences vol 35 no 6 pp1011ndash1014 1995

[15] M V Diudea ldquoPhenylenic and naphthylenic torirdquo FullerenesNanotubes and Carbon Nanostructures vol 10 no 4 pp 273ndash292 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

6 Journal of Chemistry

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 5 The structure of V-phenylenic nanotube

Theorem 7 Consider

1198601198611198622(119881119875119867119883 [119898 119899]) =

1205721 if 119898 = 119899

1205722 if 119898 = 119899

(21)

where 1205721and 120572

2are presented in the process of the proof

Proof First we note that |119881(119881119875119867119883[119898 119899])| = 6119898119899To compute the second atom-bond connectivity index of119881119875119867119883[119898 119899] we assume that 119864

1 1198642 and 119864

3are the set of all

vertical oblique and horizontal edges respectivelyThen weinfer

1198601198611198622(119881119875119867119883 [119898 119899])

= sum119890isin119864(119881119875119867119883[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(22)

Now we calculate sum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) by con-sidering the following two cases

Case 1 (119898 = 119899) Under this situation we deduce

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(23)where 119878

119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

in this case we infer that the second atom-bond connectivityindex of 119881119875119867119883[119898 119899] is

1205721= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 2|119898minus119899|minus1

sum119894=1

( ((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) )

minus1

)

12

(24)Case 2 (119898 = 119899) In this case we get

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(25)

Thus the corresponding second atom-bond connectivityindex is

1205722= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(26)The result is archived

Journal of Chemistry 7

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 6 The structure of V-phenylenic nanotorus

Our last result computes the second atom-bond connec-tivity index of 119881119875119867119884[119898 119899] and the structure of 119881119875119867119884[119898 119899]presented in Figure 6

Theorem 8 Consider

1198601198611198622(119881119875119867119884 [119898 119899]) =

1205723 if 119898 = 119899

1205724 if 119898 = 119899

(27)

where 1205723and 120572

4are presented in the process of the proof

Proof In order to prove the theorem we use a similar trickas in Theorem 7 Obviously |119881(119881119875119867119884[119898 119899])| = 6119898119899 Weassume that 119864

1 1198642 and 119864

3are the set of all vertical oblique

and horizontal edges respectively Then we deduce

1198601198611198622(119881119875119867119884 [119898 119899])

= sum119890isin119864(119881119875119867119884[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(28)

Now we calculatesum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) accord-ing to the relationship between 119898 and 119899

Case 1 (119898 = 119899) Under this assumption we infer

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ( (2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3)119894))

minus1

)

12

(29)

where 119878119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

the second atom-bond connectivity index of119881119875119867119884[119898 119899] for119898 = 119899 is

1205723= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894))

minus1

)12

(30)

Case 2 (119898 = 119899) In this case we have

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(31)

8 Journal of Chemistry

Thus the corresponding second atom-bond connectivityindex is

1205724= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(32)

The proof is completed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for their constructive com-ments in improving the quality of this paperThey also wouldlike to thank the anonymous referees for providing themwith constructive comments and suggestions The researchis partially supported by NSFC (nos 11401519 11371328 and11471293)

References

[1] L Yan Y Li W Gao and J Li ldquoPI index for some specialgraphsrdquo Journal of Chemical and Pharmaceutical Research vol5 no 11 pp 260ndash264 2013

[2] L Yan Y Li W Gao and J Li ldquoOn the extremal hyper-wienerindex of graphsrdquo Journal of Chemical and PharmaceuticalResearch vol 6 no 3 pp 477ndash481 2014

[3] W Gao and L Shi ldquoWiener index of gear fan graph and gearwheel graphrdquo Asian Journal of Chemistry vol 26 no 11 pp3397ndash3400 2014

[4] W F Xi and W Gao ldquoGeometric-arithmetic index and Zagrebindices of certain special molecular graphsrdquo Journal of Advancesin Chemistry vol 10 no 2 pp 2254ndash2261 2014

[5] J Y Dou Y Y Wang and W Gao ldquoSome characteristicson hyper-wiener index of graphsrdquo Journal of Chemical andPharmaceutical Research vol 6 no 5 pp 1659ndash1663 2014

[6] J A Bondy and U S R Murty GraphTheory Springer Berlin Germany 2008

[7] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry A vol 37 no 10 pp849ndash855 1998

[8] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquoChemical Physics Letters vol 511 pp 452ndash454 2011

[9] B Furtula A Graovac and D Vukicevic ldquoAtom-bond connec-tivity index of treesrdquo Discrete Applied Mathematics vol 157 no13 pp 2828ndash2835 2009

[10] T S Vassilev and L J Huntington ldquoOn the minimum ABCindex of chemical treesrdquo Applied Mathematics vol 2 no 1 pp8ndash16 2012

[11] J Chen J Liu and Q Li ldquoThe atom-bond connectivity index ofcatacondensed polyomino graphsrdquoDiscrete Dynamics inNatureand Society vol 2013 Article ID 598517 7 pages 2013

[12] A Graovac and M Ghorbani ldquoA new version of atom-bondconnectivity indexrdquo Acta Chimica Slovenica vol 57 no 3 pp609ndash612 2010

[13] M Rostami M S Haghighat and M Ghorbani ldquoOn secondatom-bond connectivity indexrdquo Iranian Journal of Mathemati-cal Chemistry vol 4 no 2 pp 265ndash270 2013

[14] I Gutman and S Klavzar ldquoAn algorithm for the calculationof the Szeged index of benzenoid hydrocarbonsrdquo Journal ofChemical Information and Computer Sciences vol 35 no 6 pp1011ndash1014 1995

[15] M V Diudea ldquoPhenylenic and naphthylenic torirdquo FullerenesNanotubes and Carbon Nanostructures vol 10 no 4 pp 273ndash292 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Journal of Chemistry 7

m

2

1 2 nmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Figure 6 The structure of V-phenylenic nanotorus

Our last result computes the second atom-bond connec-tivity index of 119881119875119867119884[119898 119899] and the structure of 119881119875119867119884[119898 119899]presented in Figure 6

Theorem 8 Consider

1198601198611198622(119881119875119867119884 [119898 119899]) =

1205723 if 119898 = 119899

1205724 if 119898 = 119899

(27)

where 1205723and 120572

4are presented in the process of the proof

Proof In order to prove the theorem we use a similar trickas in Theorem 7 Obviously |119881(119881119875119867119884[119898 119899])| = 6119898119899 Weassume that 119864

1 1198642 and 119864

3are the set of all vertical oblique

and horizontal edges respectively Then we deduce

1198601198611198622(119881119875119867119884 [119898 119899])

= sum119890isin119864(119881119875119867119884[119898119899])

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= sum119890isin1198641

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198642

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

(28)

Now we calculatesum119890isin1198643

radic(119899(119906) + 119899(V) minus 2)119899(119906)119899(V) accord-ing to the relationship between 119898 and 119899

Case 1 (119898 = 119899) Under this assumption we infer

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ( (2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3)119894))

minus1

)

12

(29)

where 119878119894= 3+9+15+sdot sdot sdot+(6119894minus3) and 120573 = min119898 119899 Hence

the second atom-bond connectivity index of119881119875119867119884[119898 119899] for119898 = 119899 is

1205723= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ 4|119898minus119899|minus1

sum119894=1

(((2120573) (119878120573+ (6120573 minus 3) 119894)

+ (6119898119899 minus 119878120573minus (6120573 minus 3) 119894) minus 2)

times ((2120573) (119878120573+ (6120573 minus 3) 119894)

times (6119898119899 minus 119878120573minus (6120573 minus 3) 119894))

minus1

)12

(30)

Case 2 (119898 = 119899) In this case we have

sum119890isin1198643

radic119899 (119906) + 119899 (V) minus 2

119899 (119906) 119899 (V)

= radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(31)

8 Journal of Chemistry

Thus the corresponding second atom-bond connectivityindex is

1205724= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(32)

The proof is completed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for their constructive com-ments in improving the quality of this paperThey also wouldlike to thank the anonymous referees for providing themwith constructive comments and suggestions The researchis partially supported by NSFC (nos 11401519 11371328 and11471293)

References

[1] L Yan Y Li W Gao and J Li ldquoPI index for some specialgraphsrdquo Journal of Chemical and Pharmaceutical Research vol5 no 11 pp 260ndash264 2013

[2] L Yan Y Li W Gao and J Li ldquoOn the extremal hyper-wienerindex of graphsrdquo Journal of Chemical and PharmaceuticalResearch vol 6 no 3 pp 477ndash481 2014

[3] W Gao and L Shi ldquoWiener index of gear fan graph and gearwheel graphrdquo Asian Journal of Chemistry vol 26 no 11 pp3397ndash3400 2014

[4] W F Xi and W Gao ldquoGeometric-arithmetic index and Zagrebindices of certain special molecular graphsrdquo Journal of Advancesin Chemistry vol 10 no 2 pp 2254ndash2261 2014

[5] J Y Dou Y Y Wang and W Gao ldquoSome characteristicson hyper-wiener index of graphsrdquo Journal of Chemical andPharmaceutical Research vol 6 no 5 pp 1659ndash1663 2014

[6] J A Bondy and U S R Murty GraphTheory Springer Berlin Germany 2008

[7] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry A vol 37 no 10 pp849ndash855 1998

[8] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquoChemical Physics Letters vol 511 pp 452ndash454 2011

[9] B Furtula A Graovac and D Vukicevic ldquoAtom-bond connec-tivity index of treesrdquo Discrete Applied Mathematics vol 157 no13 pp 2828ndash2835 2009

[10] T S Vassilev and L J Huntington ldquoOn the minimum ABCindex of chemical treesrdquo Applied Mathematics vol 2 no 1 pp8ndash16 2012

[11] J Chen J Liu and Q Li ldquoThe atom-bond connectivity index ofcatacondensed polyomino graphsrdquoDiscrete Dynamics inNatureand Society vol 2013 Article ID 598517 7 pages 2013

[12] A Graovac and M Ghorbani ldquoA new version of atom-bondconnectivity indexrdquo Acta Chimica Slovenica vol 57 no 3 pp609ndash612 2010

[13] M Rostami M S Haghighat and M Ghorbani ldquoOn secondatom-bond connectivity indexrdquo Iranian Journal of Mathemati-cal Chemistry vol 4 no 2 pp 265ndash270 2013

[14] I Gutman and S Klavzar ldquoAn algorithm for the calculationof the Szeged index of benzenoid hydrocarbonsrdquo Journal ofChemical Information and Computer Sciences vol 35 no 6 pp1011ndash1014 1995

[15] M V Diudea ldquoPhenylenic and naphthylenic torirdquo FullerenesNanotubes and Carbon Nanostructures vol 10 no 4 pp 273ndash292 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

8 Journal of Chemistry

Thus the corresponding second atom-bond connectivityindex is

1205724= 2119899minus1

sum119894=1

radic(6119898119894) + (6119898119899 minus 6119898119894) (2119898) minus 2

(6119898119894) (6119898119899 minus 6119898119894) (2119898)

+ 2119899minus1

sum119894=1

radic(3119898119894) + (6119898119899 minus 3119898119894) (2119899) minus 2

(3119898119894) (6119898119899 minus 3119898119894) (2119899)

+ radic4119899 (119878119899+ 6119899 minus 3) + (6119898119899 minus 119878

119899minus (6119899 minus 3)) minus 2

4119899 (119878119899+ 6119899 minus 3) (6119898119899 minus 119878

119899minus (6119899 minus 3))

(32)

The proof is completed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for their constructive com-ments in improving the quality of this paperThey also wouldlike to thank the anonymous referees for providing themwith constructive comments and suggestions The researchis partially supported by NSFC (nos 11401519 11371328 and11471293)

References

[1] L Yan Y Li W Gao and J Li ldquoPI index for some specialgraphsrdquo Journal of Chemical and Pharmaceutical Research vol5 no 11 pp 260ndash264 2013

[2] L Yan Y Li W Gao and J Li ldquoOn the extremal hyper-wienerindex of graphsrdquo Journal of Chemical and PharmaceuticalResearch vol 6 no 3 pp 477ndash481 2014

[3] W Gao and L Shi ldquoWiener index of gear fan graph and gearwheel graphrdquo Asian Journal of Chemistry vol 26 no 11 pp3397ndash3400 2014

[4] W F Xi and W Gao ldquoGeometric-arithmetic index and Zagrebindices of certain special molecular graphsrdquo Journal of Advancesin Chemistry vol 10 no 2 pp 2254ndash2261 2014

[5] J Y Dou Y Y Wang and W Gao ldquoSome characteristicson hyper-wiener index of graphsrdquo Journal of Chemical andPharmaceutical Research vol 6 no 5 pp 1659ndash1663 2014

[6] J A Bondy and U S R Murty GraphTheory Springer Berlin Germany 2008

[7] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry A vol 37 no 10 pp849ndash855 1998

[8] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquoChemical Physics Letters vol 511 pp 452ndash454 2011

[9] B Furtula A Graovac and D Vukicevic ldquoAtom-bond connec-tivity index of treesrdquo Discrete Applied Mathematics vol 157 no13 pp 2828ndash2835 2009

[10] T S Vassilev and L J Huntington ldquoOn the minimum ABCindex of chemical treesrdquo Applied Mathematics vol 2 no 1 pp8ndash16 2012

[11] J Chen J Liu and Q Li ldquoThe atom-bond connectivity index ofcatacondensed polyomino graphsrdquoDiscrete Dynamics inNatureand Society vol 2013 Article ID 598517 7 pages 2013

[12] A Graovac and M Ghorbani ldquoA new version of atom-bondconnectivity indexrdquo Acta Chimica Slovenica vol 57 no 3 pp609ndash612 2010

[13] M Rostami M S Haghighat and M Ghorbani ldquoOn secondatom-bond connectivity indexrdquo Iranian Journal of Mathemati-cal Chemistry vol 4 no 2 pp 265ndash270 2013

[14] I Gutman and S Klavzar ldquoAn algorithm for the calculationof the Szeged index of benzenoid hydrocarbonsrdquo Journal ofChemical Information and Computer Sciences vol 35 no 6 pp1011ndash1014 1995

[15] M V Diudea ldquoPhenylenic and naphthylenic torirdquo FullerenesNanotubes and Carbon Nanostructures vol 10 no 4 pp 273ndash292 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of