Research Article General Randi , Sum-Connectivity, Hyper...

Research ArticleGeneral RandiT Sum-Connectivity Hyper-Zagreb and HarmonicIndices and Harmonic Polynomial of Molecular Graphs

Mohammad Reza Farahani1 Wei Gao2 M R Rajesh Kanna3

R Pradeep Kumar4 and Jia-Bao Liu5

1Department of Applied Mathematics Iran University of Science and Technology (IUST) NarmakTehran 16844 Iran2School of Information Science and Technology Yunnan Normal University Kunming 650500 China3Department of Mathematics Maharanirsquos Science College for Women Mysore 570005 India4Department of Mathematics The National Institute of Engineering Mysuru 570008 India5School of Mathematics and Physics Anhui Jianzhu University Hefei 230601 China

Correspondence should be addressed to Mohammad Reza Farahani mrfarahani88gmailcom

Received 7 July 2016 Accepted 30 August 2016

Academic Editor Dennis Salahub

Copyright copy 2016 Mohammad Reza Farahani et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We present explicit formula for the general Randic connectivity general sum-connectivity Hyper-Zagreb and Harmonic Indicesand Harmonic polynomial of some simple connected molecular graphs

1 Introduction

In this paper we consider only simple connected graphswithout loops and multiple edges A connected graph is agraph such that there is a path between all pairs of verticesLet 119866 = (119881 119864) be an arbitrary simple connected graphwe denote the vertex set and the edge set of 119866 by 119881(119866)

and 119864(119866) respectively For two vertices 119906 and V of 119881(119866)the distance between 119906 and V is denoted by 119889(119906 V) anddefined as the length of any shortest path connecting 119906

and V in 119866 For a vertex V of 119881(119866) the degree of V isdenoted by 119889V and is the number of vertices of 119866 adjacent toV

In chemical graph theory we have many invariant poly-nomials and topological indices for a molecular graph Atopological index is a numerical value for correlation ofchemical structure with various physical properties chemicalreactivity or biological activity [1ndash3]

One of the oldest topological indices or moleculardescriptors is the Zagreb index that has been introducedmore than forty years ago by Gutman and Trinajstic in 1972[4]

Now we know that for a molecular graph 119866 = (119881 119864) thefirst Zagreb index119872

1(119866) and the secondZagreb index119872

2(119866)

are defined as

1198721(119866) = sum

Visin119881(119866)(119889V)2

or 1198721(119866) = sum

119890=119906V isin119864(119866)(119889119906+ 119889V)

1198722(119866) = sum

119890=119906V isin119864(119866)(119889119906times 119889V)

(1)

Recently a new version of Zagreb indices named Hyper-Zagreb index was introduced by Shirdel et al in 2013 [5] andit is defined as

119867119872(119866) = sum

119890=119906V isin119864(119866)(119889119906+ 119889V)2

(2)

We encourage the reader to consult [6ndash30] for historicalbackground and mathematical properties of the Zagrebindices

Hindawi Publishing CorporationAdvances in Physical ChemistryVolume 2016 Article ID 2315949 6 pageshttpdxdoiorg10115520162315949

2 Advances in Physical Chemistry

In 1975 Randic proposed a structural descriptor calledthe branching index [31] that later became the well-knownRandic molecular connectivity index Motivated by the defi-nition of Randic connectivity index based on the end-vertexdegrees of edges in a graph defined as the sum of the weights(119889119906119889V)minus12 of all edges 119906V of 119866

119877 (119866) = sum

119890=119906V isin119864(119866)

1

radic119889119906119889V

(3)

Later the Randic connectivity index had been extendedas the general Randic connectivity index which is defined asthe sum of the weights (119889

119906119889V)120572

(forall120572 isin Q) and is equal to

119877120572(119866) = sum

119890=119906V isin119864(119866)(119889119906119889V)120572

(4)

Also a closely related variant of Randic connectivityindex called the sum-connectivity index was introduced byZhou and Trinajstic in 2008 [32 33] The sum-connectivityindex 119883(119866) is defined as

119883 (119866) = sum

119906Visin119864(119866)

1

radic119889119906+ 119889V

(5)

The general sum-connectivity index of a graph 119866 is equalto (forall120572 isin Q)

119883120572(119866) = sum

119890=119906V isin119864(119866)(119889119906+ 119889V)120572

(6)

In 1987 [34] Fajtlowicz introduced the Harmonic index119867(119866) of a graph 119866 which is defined as the sum of the weights2(119889119906119889V)minus1 of forall119906V isin 119866 and is equal to

119867(119866) = sum

119890=119906Visin119864(119866)

2

119889119906+ 119889V

(7)

TheHarmonic index is one of the most important indicesin chemical and mathematical fields It is a variant of theRandic index which is themost successful molecular descrip-tor in structure-property and structure activity relationshipsstudies The Harmonic index gives somewhat better correla-tions with physical and chemical properties compared withthe well-known Randic index Estimating bounds for 119867(119866)

is of great interest and many results have been obtainedFor example Favaron et al [35] considered the relationshipbetween the Harmonic index and the eigenvalues of graphsand Zhong [36ndash38] determined theminimum andmaximumvalues of the Harmonic index for simple connected graphstrees unicyclic graphs and bicyclic graphs and characterizedthe corresponding extremal graphs respectively It turns outthat trees with maximum andminimumHarmonic index arethe path 119875

119899and the star 119878

119899 respectively

Recently Iranmanesh and Salehi [39] introduced theHarmonic polynomial 119867(119866 119909) of a graph 119866 which is equalto

119867(119866 119909) = sum

119890=119906V isin119864(119866)2119909(119889V+119889Vminus1) (8)

where 119867(119866) = int1

0119867(119866 119909)119889119909

We encourage the reader to consult [40ndash43] for morehistory and mathematical properties of the Randic index andthe Harmonic index

In this paper we present explicit formula for the gen-eral Randic connectivity general sum-connectivity Hyper-Zagreb and Harmonic Indices and Harmonic polynomial ofsome hydrocarbon molecular graphs

2 Results and Discussion

In this section we compute the general Randic connectivitygeneral sum-connectivity indices the Hyper-Zagreb andHarmonic Indices and Harmonic polynomial of a family ofhydrocarbonmolecules which are called PolycyclicAromaticHydrocarbons PAH

119896(forall119896 isin N)

The Polycyclic Aromatic Hydrocarbons PAH119896is a family

of hydrocarbonmolecules such that its structure is consistingof cycles with length six (benzene) The Polycyclic AromaticHydrocarbons can be thought of as small pieces of graphenesheets with the free valences of the dangling bonds saturatedby 119867 Vice versa a graphene sheet can be interpreted asan infinite PAH molecule Successful utilization of PAHmolecules in modeling graphite surfaces has been reportedearlier [44ndash52] and references therein Some first membersand a general representation of this hydrocarbon molecularfamily are shown in Figures 1 and 2

Theorem 1 (see [45]) Consider the Polycyclic Aromatic Hy-drocarbons PAH

119896(forall119896 isin N) Then the first and second Zagreb

indices of PAH119896are equal to

1198721(119875119860119867

119896) = 54119896

2+ 6119896

1198721(119875119860119867

119896) = 81119896

2minus 3119896

(9)

Theorem 2 The Hyper-Zagreb index of Polycyclic AromaticHydrocarbons PAH

119896(forall119896 isin N) is equal to

119867119872(PAH119896) = 12119896 (27119896 minus 1) (10)

Theorem 3 (see [46]) The Randic connectivity and sum-connectivity indices of the Polycyclic Aromatic HydrocarbonsPAH119896(forall119896 isin N) are equal to

119877 (119875119860119867119896) = 31198962 + (2radic3 minus 1) 119896

119883 (119875119860119867119896) =

119899

2(3radic6119899 + 6 minus radic6)

(11)

Advances in Physical Chemistry 3

Benzene

Coronene

Circumcoronene

Figure 1 Some first members of the Polycyclic Aromatic Hydrocarbons (PAH119896)

Figure 2 A general representation of the hydrocarbon molecularfamily ldquoPolycyclic Aromatic Hydrocarbons PAH

119896rdquo

Theorem 4 Let PAH119896be the Polycyclic Aromatic Hydrocar-

bons Then

(i) the general Randic connectivity index of PAH119896is equal

to

119877120572(119875119860119867

119896) = 3119896 (3

2120572+1119896 + 3120572(2 minus 3

120572)) (12)

(ii) the general sum-connectivity index of PAH119896is equal to

119883120572(119875119860119867

119896) = 21205723119896 (3120572+1

119896 minus 3120572+ 2120572+1

) (13)

Theorem 5 Consider the Polycyclic Aromatic HydrocarbonsPAH119896 Then

(i) the Harmonic index of PAH119896is equal to forall119896 isin N

119867(119875119860119867119896) = 3119896

2+ 2119896 (14)

(ii) the Harmonic polynomial of PAH119896is equal to forall119896 isin N

119867(119875119860119867119896 119909) = 12119896119909

3+ 6119896 (3119896 minus 1) 119909

5 (15)

Before presenting themain results consider the followingdefinition

Definition 6 (see [10]) Let119866 be a simple connectedmoleculargraph We divide the vertex set 119881(119866) and edge set 119864(119866) of 119866based on the degrees 119889V of a vertexatom V in 119866 Obviously1 le 119889V le 119899 minus 1 and we denote the minimum and maximumof the 119889V by 120575 and Δ respectively

119881119896= V isin 119881 (119866) | 119889V = 119896 forall119896 120575 le 119896 le Δ

119864119894= 119890 = 119906V isin 119864 (119866) | 119889

119906+ 119889V = 119894

forall119894 2120575 le 119894 le 2Δ

119864119895

lowast= 119906V isin 119864 (119866) | 119889

119906times 119889V = 119895 forall119895 120575

2le 119895 le Δ

2

(16)

Proof of Theorem 2 Let PAH119896be the Polycyclic Aromatic

Hydrocarbon for all integer numbers 119896 From the generalrepresentation of PAH

119896in Figure 2 one can see that in

this hydrocarbon molecular family there are 61198962+ 6119896 ver-

ticesatoms (= |119881(PAH119896)|) such that 61198962 of them are carbon

atoms and also 6119896 of them are hydrogen atoms In otherwords

1198813= 119881119862

= V isin 119881 (PAH119896) | 119889V = 3

1198811= 119881119867

= V isin 119881 (PAH119896) | 119889V = 1

(17)

Thus there are (3 times 61198962

+ 1 times 6119896)2 = 91198962

+ 3119896

edgeschemical bonds (= |119864(PAH119899)|) in PAH

119896

Now by usingDefinition 6 and according to Figure 2 onecan see that in hydrocarbon molecules PAH

119896all hydrogen

atoms have one connection and the degree of them is119889Hydrogen = 1 and there are 3 edgeschemical bonds for allcarbon atoms thus 119889Carbon = 3

Therefore we have two partitions of the vertex set119881(PAH

119899) of Polycyclic Aromatic


Hydrocarbons PAH119896are as follows

1198644= 119864lowast

3

= 119890 = 119906V = 119867119862 isin 119864 (PAH119899) | 119889119867

= 1 119889119862

= 3

1198646= 119864lowast

9

= 119890 = 119906V = 119862119862 isin 119864 (PAH119899) | 119889119862

= 119889119862

= 3

(18)

On the other hand from Figure 2 and [45 46] we can seethat |119864

4| = |119864

lowast

3| = 6119896 and |119864

6| = |119864

lowast

9| = 9119896

2minus 3119896

Here we have the following computations for the Hyper-Zagreb index of the Polycyclic Aromatic HydrocarbonsPAH119896(forall119896 isin N) as follows

119867119872(PAH119896) = sum

119890=119906Visin119864(PAH119896)(119889119906+ 119889V)2

= sum

119906Visin1198644sube119864(PAH119896)(119889119906+ 119889V)2

+ sum

119906Visin1198646sube119864(PAH119896)(119889119906+ 119889V)2

= sum

119906Visin1198644sube119864(PAH119896)(119889119862+ 119889119867)2

+ sum

119906Visin1198646sube119864(PAH119896)(119889119862+ 119889119862)2

= 16 times10038161003816100381610038161198644

1003816100381610038161003816 + 36 times10038161003816100381610038161198646

1003816100381610038161003816

= 16 times (6119896) + 36 times (91198962minus 3119896)

= 12119896 (27119896 minus 1)

(19)

Here we complete the proof of Theorem 2

Proof of Theorem 4 Consider the Polycyclic AromaticHydrocarbons PAH

119896with 6119896

2+6119896 verticesatoms and 9119896

2+3119896

edges Then by using the results from the above proof wehave the following computations for the general Randic andsum-connectivity indices of PAH

119896(forall119896 isin N forall120572 isin Q)

119877120572(PAH

119896) = sum

119890=119906Visin119864(PAH119896)(119889119906119889V)120572

= sum

119867119862isin119864lowast

3

(119889119867119889119862)120572

+ sum

119862119862isin119864lowast

9

(119889119862119889119862)120572

=1003816100381610038161003816119864lowast

3

1003816100381610038161003816 times (3)120572+

1003816100381610038161003816119864lowast

9

1003816100381610038161003816 times (9)120572

= 3120572(6119896) + 3

2120572(91198962minus 3119896)

= 3119896 (32120572+1

119896 + 3120572(2 minus 3

120572))

119883120572(PAH

119896) = sum

119890=119906Visin119864(PAH119896)(119889119906+ 119889V)120572

= sum

119867119862isin1198644

(119889119867

+ 119889119862)120572

+ sum

119862119862isin1198646

(119889119862+ 119889119862)120572

=10038161003816100381610038161198644

1003816100381610038161003816 times (4)120572+

100381610038161003816100381611986461003816100381610038161003816 times (6)

120572

= 22120572

(6119896) + 6120572(91198962minus 3119896)

= 21205723119896 (3120572+1

119896 minus 3120572+ 2120572+1

)

(20)


Hydrocarbon for all integer numbers 119896 By results from proofofTheorem 2 we see that the Harmonic index andHarmonicpolynomial of PAH

119896(forall119896 isin N) are equal to

119867(PAH119896) = sum

119906V isin119864(PAH119896)

2

119889119906+ 119889V

= sum

119906V isin119864(PAH119896)

2

119889119906+ 119889V

+ sum

119906Visin119864(PAH119896)

2

119889119906+ 119889V

= sum

119906V isin1198644

2

119889119867

+ 119889119862

+ sum

119906Visin1198646

2

119889119862+ 119889119862

=2

4

100381610038161003816100381611986441003816100381610038161003816 +

2

6

100381610038161003816100381611986461003816100381610038161003816

=2

4(6119896) +

2

6(91198962minus 3119896)

= 3119896 + (31198962minus 119896) = 3119896

2+ 2119896

119867 (PAH119896 119909) = sum

119906V isin119864(PAH119896)2119909(119889V+119889Vminus1)

=sum

119867119862isin1198644

2119909(119889119867+119889119862minus1)

+ sum

119862119862isin1198646

2119909(119889119862+119889119862minus1)

= 210038161003816100381610038161198644

1003816100381610038161003816 1199093+ 2

100381610038161003816100381611986461003816100381610038161003816 1199095

= 121198961199093+ 6119896 (3119896 minus 1) 119909

5

(21)

Here the proof of Theorem 5 was completed

Competing Interests

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

Acknowledgments

The authors are thankful to Professor Mircea V DiudeaFaculty of Chemistry and Chemical Engineering Babes-Bolyai University for his precious support and suggestions


References

[1] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947

[2] D B West Introduction to Graph Theory Prentice Hall 1996[3] R Todeschini andVConsonniHandbook ofMolecularDescrip-

tors John Wiley amp Sons Weinheim Germany 2000[4] N Trinajstic Chemical GraphTheory Mathematical Chemistry

Series CRC Press Boca Raton Fla USA 2nd edition 1992[5] I Gutman and N Trinajstic ldquoGraph theory and molecular

orbitals Total 120593-electron energy of alternant hydrocarbonsrdquoChemical Physics Letters vol 17 no 4 pp 535ndash538 1972

[6] G H Shirdel H RezaPour and A M Sayadi ldquoThe hyper-zagreb index of graph operationsrdquo Iranian Journal ofMathemat-ical Chemistry vol 4 no 2 pp 213ndash220 2013

[7] J Braun A Kerber M Meringer and C Rucker ldquoSimilarity ofmolecular descriptors the equivalence of Zagreb indices andwalk countsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 54 no 1 pp 163ndash176 2005

[8] KCDas and IGutman ldquoSomeproperties of the secondZagrebindexrdquoMATCH Communications in Mathematical and in Com-puter Chemistry vol 52 pp 103ndash112 2004

[9] T Doslic ldquoOn discriminativity of zagreb indicesrdquo IranianJournal of Mathematical Chemistry vol 3 no 1 pp 25ndash34 2012

[10] M Eliasi A Iranmanesh and I Gutman ldquoMultiplicative ver-sions of first Zagreb indexrdquoMATCH Communications in Math-ematical and in Computer Chemistry vol 68 no 1 pp 217ndash2302012

[11] M R Farahani ldquoSome connectivity indices and zagreb index ofpolyhex nanotubesrdquo Acta Chimica Slovenica vol 59 no 4 pp779ndash783 2012

[12] M R Farahani ldquoZagreb index zagreb polynomial of circum-coronene series of benzenoidrdquo Advances in Materials and Cor-rosion vol 2 no 1 pp 16ndash19 2013

[13] M R Farahani and M P Vlad ldquoComputing First and SecondZagreb index First and Second Zagreb Polynomial of Capra-designed planar benzenoid series Ca

119899(C6)rdquo Studia Universitatis

Babes-Bolyai Chemia vol 58 no 2 pp 133ndash142 2013[14] M R Farahani ldquoThe first and second zagreb indices first

and second zagreb polynomials of HAC5C6C7[pq] and

HAC5C7[pq] nanotubesrdquo International Journal of Nanoscience

and Nanotechnology vol 8 no 3 pp 175ndash180 2012[15] M R Farahani ldquoZagreb indices and Zagreb polynomials

of pent-heptagon nanotube VAC5C7(S)rdquo Chemical Physics

Research Journal vol 6 no 1 pp 35ndash40 2013[16] M R Farahani ldquoThe hyper-zagreb index of benzenoid seriesrdquo

Frontiers of Mathematics and Its Applications vol 2 no 1 pp1ndash5 2015

[17] M R Farahani ldquoComputing the hyper-zagreb index of hexag-onal nanotubesrdquo Journal of Chemistry and Materials Researchvol 2 no 1 pp 16ndash18 2015

[18] M R Farahani ldquoThe hyper-zagreb index of TUSC4C8(S)nanotubesrdquo International Journal of Engineering and TechnologyResearch vol 3 no 1 pp 1ndash6 2015

[19] I Gutman and K C Das ldquoThe first Zagreb index 30 years afterrdquoMATCH Communications in Mathematical and in ComputerChemistry no 50 pp 83ndash92 2004

[20] D Janezic A Milicevic S Nikolic N Trinajstic and D Vuk-icevic ldquoZagreb indices extension to weighted graphs represent-ing molecules containing heteroatomsrdquo Croatica Chemica Actavol 80 no 3-4 pp 541ndash545 2007

[21] S Nikolic G Kovacevic A Milicevic and N Trinajstic ldquoTheZagreb indices 30 years afterrdquo Croatica Chemica Acta vol 76no 2 pp 113ndash124 2003

[22] D Vukicevic S Nikolic and N Trinajstic ldquoOn the path-Zagrebmatrixrdquo Journal of Mathematical Chemistry vol 45 no 2 pp538ndash543 2009

[23] D Vukicevic and N Trinajstic ldquoOn the discriminatory powerof the Zagreb indices for molecular graphsrdquoMATCH Commu-nications in Mathematical and in Computer Chemistry vol 53no 1 pp 111ndash138 2005

[24] K Xu and K C Das ldquoTrees unicyclic and bicyclic graphsextremal with respect to multiplicative sum Zagreb indexrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 68 no 1 pp 257ndash272 2012

[25] B Zhou ldquoZagreb indicesrdquoMATCH Communications in Mathe-matical and in Computer Chemistry vol 52 pp 113ndash118 2004

[26] B Zhou and I Gutman ldquoFurther properties of Zagreb indicesrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 54 no 1 pp 233ndash239 2005

[27] B Zhou and I Gutman ldquoRelations between Wiener hyper-Wiener and Zagreb indicesrdquo Chemical Physics Letters vol 394no 1ndash3 pp 93ndash95 2004

[28] B Zhou andN Trinajstic ldquoSome properties of the reformulatedZagreb indicesrdquo Journal of Mathematical Chemistry vol 48 no3 pp 714ndash719 2010

[29] B Zhou and D A Stevanovic ldquoNote on Zagreb indicesrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 56 pp 571ndash578 2006

[30] B Zhou ldquoUpper bounds for the zagreb indices and the spectralradius of series-parallel graphsrdquo International Journal of Quan-tum Chemistry vol 107 no 4 pp 875ndash878 2007

[31] B Zhou ldquoRemarks on Zagreb indicesrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 57 no 3pp 591ndash596 2007

[32] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975

[33] B Zhou and N Trinajstic ldquoOn a novel connectivity indexrdquoJournal ofMathematical Chemistry vol 46 no 4 pp 1252ndash12702009

[34] B Zhou andN Trinajstic ldquoOn general sum-connectivity indexrdquoJournal of Mathematical Chemistry vol 47 no 1 pp 210ndash2182010

[35] S Fajtlowicz ldquoOn conjectures of GRAFFITI IIrdquo CongressusNumerantium vol 60 pp 187ndash197 1987

[36] O Favaron M Maheo and J F Sacle ldquoSome eigenvalueproperties in graphs (conjectures of GraffitimdashII)rdquo DiscreteMathematics vol 111 no 1ndash3 pp 197ndash220 1993

[37] L Zhong ldquoThe harmonic index for graphsrdquoApplied Mathemat-ics Letters vol 25 no 3 pp 561ndash566 2012

[38] L Zhong ldquoThe harmonic index on unicyclic graphsrdquo ArsCombinatoria vol 104 pp 261ndash269 2012

[39] L Zhong and K Xu ldquoThe harmonic index for bicyclic graphsrdquoUtilitas Mathematica vol 90 pp 23ndash32 2013

[40] M A Iranmanesh and M Salehi ldquoOn the harmonic indexand harmonic polynomial of Caterpillars with diameter fourrdquoIranian Journal of Mathematical Chemistry vol 5 no 2 pp 35ndash43 2014

[41] H Deng S Balachandran S K Ayyaswamy and Y BVenkatakrishnan ldquoOn the harmonic index and the chromatic


number of a graphrdquo Discrete Applied Mathematics vol 161 no16-17 pp 2740ndash2744 2013

[42] J Li andW C Shiu ldquoThe harmonic index of a graphrdquoTheRockyMountain Journal of Mathematics vol 44 no 5 pp 1607ndash16202014

[43] J-B Lv J Li and W C Shiu ldquoThe harmonic index of unicyclicgraphs with given matching numberrdquo Kragujevac Journal ofMathematics vol 38 no 1 pp 173ndash183 2014

[44] R Wu Z Tang and H Deng ldquoA lower bound for the harmonicindex of a graph with minimum degree at least twordquo Filomatvol 27 no 1 pp 51ndash55 2013

[45] U E Wiersum and L W Jenneskens Gas Phase Reactions inOrganic Synthesis Edited by Y Valle Gordon and BreachScience Publishers Amsterdam The Netherlands 1997

[46] M R Farahani ldquoSome connectivity indices of polycyclic aro-matic hydrocarbons PAHsrdquo Advances in Materials and Corro-sion vol 1 pp 65ndash69 2013

[47] M R Farahani ldquoZagreb indices and zagreb polynomials ofpolycyclic aromatic hydrocarbonsrdquo Journal of Chemica Actavol 2 pp 70ndash72 2013

[48] M R Farahani and W Gao ldquoOn Multiple zagreb indices ofpolycyclic aromatic hydrocarbons PAHkrdquo Journal of Chemicaland Pharmaceutical Research vol 7 no 10 pp 535ndash539 2015

[49] W Gao andM R Farahani ldquoTheTheta polynomialΘ(Gx) andthe Theta index Θ(G) of molecular graph Polycyclic AromaticHydrocarbons PAH

119896rdquo Journal of Advances in Chemistry vol 12

no 1 pp 3934ndash3939 2015[50] M R Farahani W Gao and M R Rajesh Kanna ldquoOn the

omega polynomial of a family of hydrocarbon moleculeslsquopolycyclic aromatic hydrocarbons PAHkrsquordquo Asian AcademicResearch Journal of Multidisciplinary vol 2 no 7 pp 263ndash2682015

[51] M R Farahani and M R Rajesh Kanna ldquoThe Pi polynomialand the Pi Index of a family hydrocarbons moleculesrdquo Journalof Chemical and Pharmaceutical Research vol 7 no 11 pp 253ndash257 2015

[52] M R Farahani W Gao and M R Rajesh Kanna ldquoThe edge-szeged index of the polycyclic aromatic hydrocarbons PAHkrdquoAsian Academic Research Journal of Multidisciplinary vol 2 no7 pp 136ndash142 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CatalystsJournal of


In 1975 Randic proposed a structural descriptor calledthe branching index [31] that later became the well-knownRandic molecular connectivity index Motivated by the defi-nition of Randic connectivity index based on the end-vertexdegrees of edges in a graph defined as the sum of the weights(119889119906119889V)minus12 of all edges 119906V of 119866

119877 (119866) = sum

119890=119906V isin119864(119866)

1

radic119889119906119889V

(3)

Later the Randic connectivity index had been extendedas the general Randic connectivity index which is defined asthe sum of the weights (119889

119906119889V)120572

(forall120572 isin Q) and is equal to

119877120572(119866) = sum

119890=119906V isin119864(119866)(119889119906119889V)120572

(4)

Also a closely related variant of Randic connectivityindex called the sum-connectivity index was introduced byZhou and Trinajstic in 2008 [32 33] The sum-connectivityindex 119883(119866) is defined as

119883 (119866) = sum

119906Visin119864(119866)

1

radic119889119906+ 119889V

(5)

The general sum-connectivity index of a graph 119866 is equalto (forall120572 isin Q)

119883120572(119866) = sum

119890=119906V isin119864(119866)(119889119906+ 119889V)120572

(6)

In 1987 [34] Fajtlowicz introduced the Harmonic index119867(119866) of a graph 119866 which is defined as the sum of the weights2(119889119906119889V)minus1 of forall119906V isin 119866 and is equal to

119867(119866) = sum

119890=119906Visin119864(119866)

2

119889119906+ 119889V

(7)

TheHarmonic index is one of the most important indicesin chemical and mathematical fields It is a variant of theRandic index which is themost successful molecular descrip-tor in structure-property and structure activity relationshipsstudies The Harmonic index gives somewhat better correla-tions with physical and chemical properties compared withthe well-known Randic index Estimating bounds for 119867(119866)

is of great interest and many results have been obtainedFor example Favaron et al [35] considered the relationshipbetween the Harmonic index and the eigenvalues of graphsand Zhong [36ndash38] determined theminimum andmaximumvalues of the Harmonic index for simple connected graphstrees unicyclic graphs and bicyclic graphs and characterizedthe corresponding extremal graphs respectively It turns outthat trees with maximum andminimumHarmonic index arethe path 119875

119899and the star 119878

119899 respectively

Recently Iranmanesh and Salehi [39] introduced theHarmonic polynomial 119867(119866 119909) of a graph 119866 which is equalto

119867(119866 119909) = sum

119890=119906V isin119864(119866)2119909(119889V+119889Vminus1) (8)

where 119867(119866) = int1

0119867(119866 119909)119889119909

We encourage the reader to consult [40ndash43] for morehistory and mathematical properties of the Randic index andthe Harmonic index

In this paper we present explicit formula for the gen-eral Randic connectivity general sum-connectivity Hyper-Zagreb and Harmonic Indices and Harmonic polynomial ofsome hydrocarbon molecular graphs

2 Results and Discussion

In this section we compute the general Randic connectivitygeneral sum-connectivity indices the Hyper-Zagreb andHarmonic Indices and Harmonic polynomial of a family ofhydrocarbonmolecules which are called PolycyclicAromaticHydrocarbons PAH

119896(forall119896 isin N)

The Polycyclic Aromatic Hydrocarbons PAH119896is a family

of hydrocarbonmolecules such that its structure is consistingof cycles with length six (benzene) The Polycyclic AromaticHydrocarbons can be thought of as small pieces of graphenesheets with the free valences of the dangling bonds saturatedby 119867 Vice versa a graphene sheet can be interpreted asan infinite PAH molecule Successful utilization of PAHmolecules in modeling graphite surfaces has been reportedearlier [44ndash52] and references therein Some first membersand a general representation of this hydrocarbon molecularfamily are shown in Figures 1 and 2

Theorem 1 (see [45]) Consider the Polycyclic Aromatic Hy-drocarbons PAH

119896(forall119896 isin N) Then the first and second Zagreb

indices of PAH119896are equal to

1198721(119875119860119867

119896) = 54119896

2+ 6119896

1198721(119875119860119867

119896) = 81119896

2minus 3119896

(9)

Theorem 2 The Hyper-Zagreb index of Polycyclic AromaticHydrocarbons PAH

119896(forall119896 isin N) is equal to

119867119872(PAH119896) = 12119896 (27119896 minus 1) (10)

Theorem 3 (see [46]) The Randic connectivity and sum-connectivity indices of the Polycyclic Aromatic HydrocarbonsPAH119896(forall119896 isin N) are equal to

119877 (119875119860119867119896) = 31198962 + (2radic3 minus 1) 119896

119883 (119875119860119867119896) =

119899

2(3radic6119899 + 6 minus radic6)

(11)


Benzene

Coronene

Circumcoronene



119896rdquo


bons Then


to

119877120572(119875119860119867

119896) = 3119896 (3

2120572+1119896 + 3120572(2 minus 3

120572)) (12)


119883120572(119875119860119867

119896) = 21205723119896 (3120572+1

119896 minus 3120572+ 2120572+1

) (13)



119867(119875119860119867119896) = 3119896

2+ 2119896 (14)


119867(119875119860119867119896 119909) = 12119896119909

3+ 6119896 (3119896 minus 1) 119909

5 (15)




119864119894= 119890 = 119906V isin 119864 (119866) | 119889

119906+ 119889V = 119894

forall119894 2120575 le 119894 le 2Δ

119864119895

lowast= 119906V isin 119864 (119866) | 119889

119906times 119889V = 119895 forall119895 120575

2le 119895 le Δ

2

(16)







1198813= 119881119862

= V isin 119881 (PAH119896) | 119889V = 3

1198811= 119881119867

= V isin 119881 (PAH119896) | 119889V = 1

(17)


+ 1 times 6119896)2 = 91198962

+ 3119896


119896


119896all hydrogen






1198644= 119864lowast

3

= 119890 = 119906V = 119867119862 isin 119864 (PAH119899) | 119889119867

= 1 119889119862

= 3

1198646= 119864lowast

9

= 119890 = 119906V = 119862119862 isin 119864 (PAH119899) | 119889119862

= 119889119862

= 3

(18)


4| = |119864

lowast

3| = 6119896 and |119864

6| = |119864

lowast

9| = 9119896

2minus 3119896


119867119872(PAH119896) = sum

119890=119906Visin119864(PAH119896)(119889119906+ 119889V)2

= sum

119906Visin1198644sube119864(PAH119896)(119889119906+ 119889V)2

+ sum

119906Visin1198646sube119864(PAH119896)(119889119906+ 119889V)2

= sum

119906Visin1198644sube119864(PAH119896)(119889119862+ 119889119867)2

+ sum

119906Visin1198646sube119864(PAH119896)(119889119862+ 119889119862)2

= 16 times10038161003816100381610038161198644

1003816100381610038161003816 + 36 times10038161003816100381610038161198646

1003816100381610038161003816


= 12119896 (27119896 minus 1)

(19)



119896with 6119896


2+3119896



119877120572(PAH

119896) = sum

119890=119906Visin119864(PAH119896)(119889119906119889V)120572

= sum

119867119862isin119864lowast

3

(119889119867119889119862)120572

+ sum

119862119862isin119864lowast

9

(119889119862119889119862)120572

=1003816100381610038161003816119864lowast

3

1003816100381610038161003816 times (3)120572+

1003816100381610038161003816119864lowast

9

1003816100381610038161003816 times (9)120572

= 3120572(6119896) + 3

2120572(91198962minus 3119896)

= 3119896 (32120572+1

119896 + 3120572(2 minus 3

120572))

119883120572(PAH

119896) = sum

119890=119906Visin119864(PAH119896)(119889119906+ 119889V)120572

= sum

119867119862isin1198644

(119889119867

+ 119889119862)120572

+ sum

119862119862isin1198646

(119889119862+ 119889119862)120572

=10038161003816100381610038161198644

1003816100381610038161003816 times (4)120572+

100381610038161003816100381611986461003816100381610038161003816 times (6)

120572

= 22120572

(6119896) + 6120572(91198962minus 3119896)

= 21205723119896 (3120572+1

119896 minus 3120572+ 2120572+1

)

(20)




119867(PAH119896) = sum

119906V isin119864(PAH119896)

2

119889119906+ 119889V

= sum

119906V isin119864(PAH119896)

2

119889119906+ 119889V

+ sum

119906Visin119864(PAH119896)

2

119889119906+ 119889V

= sum

119906V isin1198644

2

119889119867

+ 119889119862

+ sum

119906Visin1198646

2

119889119862+ 119889119862

=2

4

100381610038161003816100381611986441003816100381610038161003816 +

2

6

100381610038161003816100381611986461003816100381610038161003816

=2

4(6119896) +

2

6(91198962minus 3119896)

= 3119896 + (31198962minus 119896) = 3119896

2+ 2119896

119867 (PAH119896 119909) = sum


=sum

119867119862isin1198644

2119909(119889119867+119889119862minus1)

+ sum

119862119862isin1198646

2119909(119889119862+119889119862minus1)

= 210038161003816100381610038161198644

1003816100381610038161003816 1199093+ 2

100381610038161003816100381611986461003816100381610038161003816 1199095

= 121198961199093+ 6119896 (3119896 minus 1) 119909

5

(21)


Competing Interests


Acknowledgments



References







































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Benzene

Coronene

Circumcoronene



119896rdquo


bons Then


to

119877120572(119875119860119867

119896) = 3119896 (3

2120572+1119896 + 3120572(2 minus 3

120572)) (12)


119883120572(119875119860119867

119896) = 21205723119896 (3120572+1

119896 minus 3120572+ 2120572+1

) (13)



119867(119875119860119867119896) = 3119896

2+ 2119896 (14)


119867(119875119860119867119896 119909) = 12119896119909

3+ 6119896 (3119896 minus 1) 119909

5 (15)




119864119894= 119890 = 119906V isin 119864 (119866) | 119889

119906+ 119889V = 119894

forall119894 2120575 le 119894 le 2Δ

119864119895

lowast= 119906V isin 119864 (119866) | 119889

119906times 119889V = 119895 forall119895 120575

2le 119895 le Δ

2

(16)







1198813= 119881119862

= V isin 119881 (PAH119896) | 119889V = 3

1198811= 119881119867

= V isin 119881 (PAH119896) | 119889V = 1

(17)


+ 1 times 6119896)2 = 91198962

+ 3119896


119896


119896all hydrogen






1198644= 119864lowast

3

= 119890 = 119906V = 119867119862 isin 119864 (PAH119899) | 119889119867

= 1 119889119862

= 3

1198646= 119864lowast

9

= 119890 = 119906V = 119862119862 isin 119864 (PAH119899) | 119889119862

= 119889119862

= 3

(18)


4| = |119864

lowast

3| = 6119896 and |119864

6| = |119864

lowast

9| = 9119896

2minus 3119896


119867119872(PAH119896) = sum

119890=119906Visin119864(PAH119896)(119889119906+ 119889V)2

= sum

119906Visin1198644sube119864(PAH119896)(119889119906+ 119889V)2

+ sum

119906Visin1198646sube119864(PAH119896)(119889119906+ 119889V)2

= sum

119906Visin1198644sube119864(PAH119896)(119889119862+ 119889119867)2

+ sum

119906Visin1198646sube119864(PAH119896)(119889119862+ 119889119862)2

= 16 times10038161003816100381610038161198644

1003816100381610038161003816 + 36 times10038161003816100381610038161198646

1003816100381610038161003816


= 12119896 (27119896 minus 1)

(19)



119896with 6119896


2+3119896



119877120572(PAH

119896) = sum

119890=119906Visin119864(PAH119896)(119889119906119889V)120572

= sum

119867119862isin119864lowast

3

(119889119867119889119862)120572

+ sum

119862119862isin119864lowast

9

(119889119862119889119862)120572

=1003816100381610038161003816119864lowast

3

1003816100381610038161003816 times (3)120572+

1003816100381610038161003816119864lowast

9

1003816100381610038161003816 times (9)120572

= 3120572(6119896) + 3

2120572(91198962minus 3119896)

= 3119896 (32120572+1

119896 + 3120572(2 minus 3

120572))

119883120572(PAH

119896) = sum

119890=119906Visin119864(PAH119896)(119889119906+ 119889V)120572

= sum

119867119862isin1198644

(119889119867

+ 119889119862)120572

+ sum

119862119862isin1198646

(119889119862+ 119889119862)120572

=10038161003816100381610038161198644

1003816100381610038161003816 times (4)120572+

100381610038161003816100381611986461003816100381610038161003816 times (6)

120572

= 22120572

(6119896) + 6120572(91198962minus 3119896)

= 21205723119896 (3120572+1

119896 minus 3120572+ 2120572+1

)

(20)




119867(PAH119896) = sum

119906V isin119864(PAH119896)

2

119889119906+ 119889V

= sum

119906V isin119864(PAH119896)

2

119889119906+ 119889V

+ sum

119906Visin119864(PAH119896)

2

119889119906+ 119889V

= sum

119906V isin1198644

2

119889119867

+ 119889119862

+ sum

119906Visin1198646

2

119889119862+ 119889119862

=2

4

100381610038161003816100381611986441003816100381610038161003816 +

2

6

100381610038161003816100381611986461003816100381610038161003816

=2

4(6119896) +

2

6(91198962minus 3119896)

= 3119896 + (31198962minus 119896) = 3119896

2+ 2119896

119867 (PAH119896 119909) = sum


=sum

119867119862isin1198644

2119909(119889119867+119889119862minus1)

+ sum

119862119862isin1198646

2119909(119889119862+119889119862minus1)

= 210038161003816100381610038161198644

1003816100381610038161003816 1199093+ 2

100381610038161003816100381611986461003816100381610038161003816 1199095

= 121198961199093+ 6119896 (3119896 minus 1) 119909

5

(21)


Competing Interests


Acknowledgments



References







































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



1198644= 119864lowast

3

= 119890 = 119906V = 119867119862 isin 119864 (PAH119899) | 119889119867

= 1 119889119862

= 3

1198646= 119864lowast

9

= 119890 = 119906V = 119862119862 isin 119864 (PAH119899) | 119889119862

= 119889119862

= 3

(18)


4| = |119864

lowast

3| = 6119896 and |119864

6| = |119864

lowast

9| = 9119896

2minus 3119896


119867119872(PAH119896) = sum

119890=119906Visin119864(PAH119896)(119889119906+ 119889V)2

= sum

119906Visin1198644sube119864(PAH119896)(119889119906+ 119889V)2

+ sum

119906Visin1198646sube119864(PAH119896)(119889119906+ 119889V)2

= sum

119906Visin1198644sube119864(PAH119896)(119889119862+ 119889119867)2

+ sum

119906Visin1198646sube119864(PAH119896)(119889119862+ 119889119862)2

= 16 times10038161003816100381610038161198644

1003816100381610038161003816 + 36 times10038161003816100381610038161198646

1003816100381610038161003816


= 12119896 (27119896 minus 1)

(19)



119896with 6119896


2+3119896



119877120572(PAH

119896) = sum

119890=119906Visin119864(PAH119896)(119889119906119889V)120572

= sum

119867119862isin119864lowast

3

(119889119867119889119862)120572

+ sum

119862119862isin119864lowast

9

(119889119862119889119862)120572

=1003816100381610038161003816119864lowast

3

1003816100381610038161003816 times (3)120572+

1003816100381610038161003816119864lowast

9

1003816100381610038161003816 times (9)120572

= 3120572(6119896) + 3

2120572(91198962minus 3119896)

= 3119896 (32120572+1

119896 + 3120572(2 minus 3

120572))

119883120572(PAH

119896) = sum

119890=119906Visin119864(PAH119896)(119889119906+ 119889V)120572

= sum

119867119862isin1198644

(119889119867

+ 119889119862)120572

+ sum

119862119862isin1198646

(119889119862+ 119889119862)120572

=10038161003816100381610038161198644

1003816100381610038161003816 times (4)120572+

100381610038161003816100381611986461003816100381610038161003816 times (6)

120572

= 22120572

(6119896) + 6120572(91198962minus 3119896)

= 21205723119896 (3120572+1

119896 minus 3120572+ 2120572+1

)

(20)




119867(PAH119896) = sum

119906V isin119864(PAH119896)

2

119889119906+ 119889V

= sum

119906V isin119864(PAH119896)

2

119889119906+ 119889V

+ sum

119906Visin119864(PAH119896)

2

119889119906+ 119889V

= sum

119906V isin1198644

2

119889119867

+ 119889119862

+ sum

119906Visin1198646

2

119889119862+ 119889119862

=2

4

100381610038161003816100381611986441003816100381610038161003816 +

2

6

100381610038161003816100381611986461003816100381610038161003816

=2

4(6119896) +

2

6(91198962minus 3119896)

= 3119896 + (31198962minus 119896) = 3119896

2+ 2119896

119867 (PAH119896 119909) = sum


=sum

119867119862isin1198644

2119909(119889119867+119889119862minus1)

+ sum

119862119862isin1198646

2119909(119889119862+119889119862minus1)

= 210038161003816100381610038161198644

1003816100381610038161003816 1199093+ 2

100381610038161003816100381611986461003816100381610038161003816 1199095

= 121198961199093+ 6119896 (3119896 minus 1) 119909

5

(21)


Competing Interests


Acknowledgments



References







































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


References







































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

Research Article General Randi , Sum-Connectivity, Hyper...

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