Research Article Some Properties on Estrada Index of ...
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Research Article Some Properties on Estrada Index of Folded Hypercubes Networks Jia-Bao Liu, 1,2,3 Xiang-Feng Pan, 1 and Jinde Cao 2,4 1 School of Mathematics Science, Anhui University, Hefei 230601, China 2 Department of Mathematics, Southeast University, Nanjing 210096, China 3 Anhui Xinhua University, Hefei 230088, China 4 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Jinde Cao; [email protected] Received 24 November 2013; Accepted 19 December 2013; Published 5 February 2014 Academic Editor: Qiankun Song Copyright © 2014 Jia-Bao Liu et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let be a simple graph with vertices and let 1 , 2 ,..., be the eigenvalues of its adjacency matrix; the Estrada index () of the graph is defined as the sum of the terms , = 1,2,...,. e -dimensional folded hypercube networks are an important and attractive variant of the -dimensional hypercube networks , which are obtained from by adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networks by deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networks are proposed. 1. Introduction Complex networks have become an important area of multidisciplinary research involving mathematics, physics, social sciences, biology, and other theoretical and applied sciences. It is well known that interconnection networks play an important role in parallel communication systems. An interconnection network is usually modelled by a connected graph = (,), where denotes the set of processors and denotes the set of communication links between processors in networks. Let be a graph with vertices labelled 1, 2, . . . , . e adjacency matrix () of is an × matrix with the (, )-entry equal to 1 if vertices and are adjacent and 0 otherwise. e spectrum of is the spectrum of its adjacency matrix and consists of the numbers 1 ≥ 2 ≥ ⋅⋅⋅ ≥ . In this work we are concerned with finite undirected connected simple graphs (networks). For the underlying graph theoretical definitions and notations we follow [1]. e energy of the graph [2] is defined as () = ∑ =1 . (1) Another graph-spectrum-based invariant, recently put forward by Ernesto Estrada, is defined as = () = ∑ =1 . (2) is graph invariant appeared for the first time in the year 2000, in a paper by Estrada [3], dealing with the folding of protein molecules. Estrada and Rodr´ ıguez-Vel´ azquez showed that provides a measure of the centrality of complex (communication, social, metabolic, etc.) networks [4, 5]. Denote by = () = ∑ =1 ( ) the th spectral moment of the graph . From the Taylor expansion of , we have the following important relation between the Estrada index and the spectral moments of : () = ∞ ∑ =0 () ! . (3) At this point one should recall [4] that () is equal to the number of self-returning walks of length of the graph Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 167623, 6 pages http://dx.doi.org/10.1155/2014/167623
Transcript of Research Article Some Properties on Estrada Index of ...
Research ArticleSome Properties on Estrada Index of FoldedHypercubes Networks
Jia-Bao Liu123 Xiang-Feng Pan1 and Jinde Cao24
1 School of Mathematics Science Anhui University Hefei 230601 China2Department of Mathematics Southeast University Nanjing 210096 China3 Anhui Xinhua University Hefei 230088 China4Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia
Correspondence should be addressed to Jinde Cao jdcaoseueducn
Received 24 November 2013 Accepted 19 December 2013 Published 5 February 2014
Academic Editor Qiankun Song
Copyright copy 2014 Jia-Bao Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Let 119866 be a simple graph with 119899 vertices and let 1205821 1205822 120582
119899be the eigenvalues of its adjacency matrix the Estrada index 119864119864(119866)
of the graph 119866 is defined as the sum of the terms 119890120582119894 119894 = 1 2 119899 The 119899-dimensional folded hypercube networks 119865119876119899are an
important and attractive variant of the 119899-dimensional hypercube networks 119876119899 which are obtained from 119876
119899by adding an edge
between any pair of vertices complementary edges In this paper we establish the explicit formulae for calculating the Estradaindex of the folded hypercubes networks 119865119876
119899by deducing the characteristic polynomial of the adjacency matrix in spectral graph
theory Moreover some lower and upper bounds for the Estrada index of the folded hypercubes networks 119865119876119899are proposed
1 Introduction
Complex networks have become an important area ofmultidisciplinary research involving mathematics physicssocial sciences biology and other theoretical and appliedsciences It is well known that interconnection networks playan important role in parallel communication systems Aninterconnection network is usually modelled by a connectedgraph 119866 = (119881 119864) where 119881 denotes the set of processors and119864 denotes the set of communication links between processorsin networks Let119866 be a graphwith vertices labelled 1 2 119899The adjacency matrix 119860(119866) of 119866 is an 119899 times 119899 matrix with the(119894 119895)-entry equal to 1 if vertices 119894 and 119895 are adjacent and 0otherwiseThe spectrum of119866 is the spectrum of its adjacencymatrix and consists of the numbers 120582
1ge 1205822
ge sdot sdot sdot ge
120582119899 In this work we are concerned with finite undirected
connected simple graphs (networks) For the underlyinggraph theoretical definitions and notations we follow [1]
The energy of the graph 119866 [2] is defined as
119864 (119866) =
119899
sum
119894=1
10038161003816100381610038161205821198941003816100381610038161003816 (1)
Another graph-spectrum-based invariant recently putforward by Ernesto Estrada is defined as
119864119864 = 119864119864 (119866) =
119899
sum
119894=1
119890120582119894 (2)
This graph invariant appeared for the first time in the year2000 in a paper by Estrada [3] dealing with the folding ofprotein molecules Estrada and Rodrıguez-Velazquez showedthat 119864119864 provides a measure of the centrality of complex(communication social metabolic etc) networks [4 5]
Denote by 119872119896= 119872119896(119866) = sum
119899
119894=1(120582119894)119896 the 119896th spectral
moment of the graph 119866 From the Taylor expansion of 119890119909we have the following important relation between the Estradaindex and the spectral moments of 119866
119864119864 (119866) =
infin
sum
119896=0
119872119896(119866)
119896 (3)
At this point one should recall [4] that119872119896(119866) is equal to
the number of self-returning walks of length 119896 of the graph
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 167623 6 pageshttpdxdoiorg1011552014167623
2 Abstract and Applied Analysis
119866 The first few spectral moments of an (119899119898)-graph with 119898edges and 119905 triangles satisfy the following relations [4]
1198720=
119899
sum
119894=1
(120582119894)0
= 119899 1198721=
119899
sum
119894=1
(120582119894)1
= 0
1198722=
119899
sum
119894=1
(120582119894)2
= 2119898 1198723=
119899
sum
119894=1
(120582119894)3
= 6119905
(4)
For 1 le 119894 le 119899 let119889119894be the degree of vertex V
119894in119866The first
Zagreb index [6] of the graph119866 is defined as119885119892(119866) = sum119899
119894=11198892
119894
1198724=
119899
sum
119894=1
(120582119894)4
= 2119885119892 (119866) minus 2119898 + 8119902
1198725=
119899
sum
119894=1
(120582119894)5
= 30119905 + 10119901 + 10119903
(5)
where 119901 and 119902 are the numbers of pentagons and quadranglesin119866 and 119903 is the number of subgraphs consisting of a trianglewith a pendent vertex attached [7]
The hypercubes 119876119899is one of the most popular and
efficient interconnection networks due to its many excellentperformances for some practical applications There is alarge amount of literature on the properties of hypercubesnetworks [8ndash11] As an important variant of 119876
119899 the folded
hypercubes networks 119865119876119899 proposed by Amawy and Latifi
[8] are the graphs obtained from 119876119899by adding an edge
between any pair of vertices complementary addresses Thefolded hypercubes 119865119876
119899obtained considerable attention due
to its perfect properties such as symmetry regular structurestrong connectivity small diameter and many of its proper-ties which have been explored [12ndash19]
The remainder of the present paper is organized asfollows In Section 2 we present some basic notations andsome preliminaries in our discussionThe proofs of our mainresults are in Section 3 and some conclusions are given inSection 4 respectively
2 Notations and Some Preliminaries
In this section we introduce some basic properties which willbe used in the proofs of our main results
Let 119875119865119876119899
(119909) be the characteristic polynomial of the adja-cency matrix of the folded hypercube 119865119876
119899 the following
results were shown in [12]
Lemma 1 (see [12]) The characteristic polynomial of theadjacency matrix of the 119865119876
119899(119899 ge 3) is
119875 (119865119876119899 120582) = [120582 minus (119899 minus 7)] [120582 minus (119899 minus 3)]
3119875 (119865119876
119899minus1 120582 minus 1)
times
119899minus2
prod
119894=2
119875 (119865119876119899minus119894 120582 minus (119894 minus 4))
(6)
Lemma 2 (see [12]) For 119865119876119899with 119899 ge 3 the spectrum of
adjacency matrix is as follows(1) If 119899 equiv 0 (mod 2)
119878119901119890119888 (119865119876119899) = (
minus119899 + 1 minus119899 + 5 minus119899 + 9 sdot sdot sdot 119899 minus 7 119899 minus 3 119899 + 1
1198620
119899+ 1198621
1198991198622
119899+ 1198623
1198991198624
119899+ 1198625
119899sdot sdot sdot 119862
119899minus4
119899+ 119862119899minus3
119899119862119899minus2
119899+ 119862119899minus1
119899119862119899
119899
) (7)
(2) if 119899 equiv 1 (mod 2)
119878119901119890119888 (119865119876119899) = (
minus119899 minus 1 minus119899 + 3 minus119899 + 7 sdot sdot sdot 119899 minus 7 119899 minus 3 119899 + 1
1198620
1198991198621
119899+ 1198622
1198991198623
119899+ 1198624
119899sdot sdot sdot 119862
119899minus4
119899+ 119862119899minus3
119899119862119899minus2
119899+ 119862119899minus1
119899119862119899
119899
) (8)
where 119862119894119899are the binomial coefficients and the elements in
the first and second rows are the eigenvalues of the adjacencymatrix of 119865119876
119899and the corresponding multiplicities respec-
tively
Lemma 3 (see [20]) The eigenvalues of a bipartite graphsatisfy the pairing property 120582
119899minus119894+1= 120582119894 119894 = 1 2 119899
Therefore if the graph 119866 is bipartite and if 1205780is nullity (the
multiplicity of its eigenvalue zero) then
119864119864 (119866) = 1205780+ 2sum
+
cosh (119894) (9)
where cosh stands for the hyperbolic cosine cosh(119909) =
(119890119909+ 119890minus119909)2 whereas sum
+denotes summation over all positive
eigenvalues of the corresponding graph
Lemma 4 (see [21]) Let119866 be a graph with119898 edges For 119896 ge 4119872119896+2
ge 119872119896 (10)
with equality for all even 119896 ge 4 if and only if 119866 consists of 119898copies of119870
2and possibly isolated vertices and with equality for
all odd 119896 ge 5 if and only if 119866 is a bipartite graph
The following lemma is an immediate result of theprevious lemma
Abstract and Applied Analysis 3
Lemma 5 (see [22]) Let 119866 be an (119899119898) graph with 119898 edgesFor 119896 ge 4
119899
sum
119894=1
(2120582119894)119896+2
ge 4
119899
sum
119894=1
(2120582119894)119896
(11)
with equality for all even 119896 ge 4 if and only if 119866 consists of 119898copies of119870
2and possibly isolated vertices and with equality for
all odd 119896 ge 5 if and only if 119866 is a bipartite graph
Lemma 6 (see [23]) Let 119866 be a regular graph of degree 119903 = 0
and of order 119899 Then its Estrada index is bounded by
119890119903+ (119899 minus 1) 119890
minus119903(119899minus1)le 119864119864 (119866) lt 119899 minus 2 + 119890
119903+ 119890radic119903(119899minus119903)minus1
(12)
Equality holds if and only if 1205822= 1205823= sdot sdot sdot = 120582
119899= minus119903(119899 minus 1)
Lemma 7 (see [23]) The Estrada index 119864119864(119866) and the graphenergy 119864(119866) satisfy the following inequality
1
2119864 (119866) (119890 minus 1) + 119899 minus 119899
+le 119864119864 (119866) le 119899 minus 1 + 119890
119864(119866)2 (13)
and equalities on both sides hold if and only if 119864(119866) = 0
3 Main Results
31The Estrada Index of FoldedHypercubes Networks119865119876119899 In
this section we present some explicit formulae for calculatingthe Estrada index of 119865119876
119899 For convenience we assume that
119862119894
119899= 0 if 119894 lt 0 or 119894 gt 119899
Theorem 8 For any 119865119876119899with 119899 ge 3 then
(1) 119864119864(119865119876119899) = sum
1198992
119894=0(1198622119894
119899+ 1198622119894+1
119899)1198904119894minus119899+1 119894 = 0 1 1198992
if 119899 equiv 0 (mod 2)
(2) 119864119864(119865119876119899) = sum1198992
119894=0(1198622119894minus1
119899+ 1198622119894
119899)1198904119894minus119899minus1 119894 = 0 1 (119899 +
1)2 if 119899 equiv 1 (mod 2)
where the 4119894 minus 119899 + 1 and 4119894 minus 119899 minus 1 (119894 = 0 1 1198992 or (119899 +1)2) are the eigenvalues of the adjacent matrix of 119865119876
119899and 119862119894
119899
denotes the binomial coefficients
Proof By Lemma 1 the characteristic polynomial of the ad-jacent matrix of 119865119876
119899is
119875 (119865119876119899 120582) = [120582 minus (119899 minus 7)] [120582 minus (119899 minus 3)]
3119875 (119865119876
119899minus1 120582 minus 1)
times
119899minus2
prod
119894=2
119875 (119865119876119899minus119894 120582 minus (119894 minus 4))
(14)
Through calculating eigenvalues of characteristic polyno-mial and its multiplicities we obtained that
(1) if 119899 equiv 0 (mod 2) 119865119876119899have 1198992 + 1 different eigen-
values 4119894 minus 119899 + 1 with the multiplicities 1198622119894119899+ 1198622119894+1
119899
where 119894 = 0 1 1198992
(2) if 119899 equiv 1 (mod 2) 119865119876119899have (119899 + 1)2 different
eigenvalues 4119894 minus 119899 minus 1 with the multiplicities 1198622119894minus1119899
+
1198622119894
119899 where 119894 = 0 1 (119899 + 1)2
Combining with the definition of the Estrada index wederived the result of Theorem 8
32 Some Bounds for the Estrada Index of Folded HypercubesNetworks 119865119876
119899 It is well known that 119865119876
119899have 2119899 vertices
Let 1205821ge 1205822ge sdot sdot sdot ge 120582
119899ge 120582119899+1
ge sdot sdot sdot ge 1205822119899 be the
eigenvalues of 119865119876119899with nonincreasing order In order to
obtain the bounds for the Estrada index of 119865119876119899 we prove
some results by utilizing the arithmetic and geometric meaninequality in our proof some techniques in [22] are referredto
Theorem 9 For any 119865119876119899with 119899 ge 2 one has
radic4119899+(119899 + 1) 2119899+1+8119905+[cosh (2)minus3]1198724+[cosh (2)minus10
3]1198725
lt 119864119864 (119865119876119899)
(15)
where 1198724= 2119885119892(119866) minus 2119898 + 8119902 119872
5= 30119905 + 10119901 + 10119903 119901
and 119902 are the numbers of pentagons and quadrangles in 119865119876119899
and 119903 is the number of subgraphs consisting of a triangle witha pendent vertex attached
Proof In order to obtain the lower bounds for the Estradaindex consider that
1198641198642(119865119876119899) =
2119899
sum
119894=1
1198902120582119894 + 2sum
119894lt119895
119890120582119894119890120582119895 (16)
Noting that 1198720= 2119899 1198721= 0 119872
2= (119899 + 1)2
119899minus1 and1198723= 6119905 we obtain
2119899
sum
119894=1
1198902120582119894 =
2119899
sum
119894=1
sum
119896ge0
(2120582119894)119896
119896
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 +
2119899
sum
119894=1
sum
119896ge4
(2120582119894)119896
119896
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 + sum
119896ge2
1
(2119896)
2119899
sum
119894=1
(2120582119894)2119896
+ sum
119896ge2
1
(2119896 + 1)
2119899
sum
119894=1
(2120582119894)2119896+1
(17)
By Lemma 5
119899
sum
119894=1
(2120582119894)119896+2
ge 4
119899
sum
119894=1
(2120582119894)119896
(18)
4 Abstract and Applied Analysis
we can get that
2119899
sum
119894=1
1198902120582119894 ge 2
119899+ (119899 + 1) 2
119899+1+ 8119905
+ sum
119896ge2
1
(2119896)
2119899
sum
119894=1
22119896minus4
(2120582119894)4
+ sum
119896ge2
1
(2119896 + 1)
2119899
sum
119894=1
22119896minus4
(2120582119894)5
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 + [cosh (2) minus 3]1198724
+ [cosh (2) minus 10
3]1198725
(19)
where1198724= 2119885119892(119866)minus 2119898+8119902 119872
5= 30119905 + 10119901+10119903 119901 and
119902 are the numbers of pentagons and quadrangles in 119865119876119899 and
119903 is the number of subgraphs consisting of a triangle with apendent vertex attached
As for the terms 2sum119894lt119895
119890120582119894119890120582119895 by the arithmetic and
geometric mean inequality and the fact that1198721= 0
2sum
119894lt119895
119890120582119894119890120582119895 ge 2
119899(2119899minus 1)(prod
119894lt119895
119890120582119894119890120582119895)
22119899(2119899minus1)
= 2119899(2119899minus 1)[
[
(prod
119894=1
119890120582119894)
2119899minus1
]
]
22119899(2119899minus1)
= 2119899(2119899minus 1) (119890
1198721)22119899
= 2119899(2119899minus 1)
(20)
where the equality holds if and only if 1205821= sdot sdot sdot = 120582
2119899
Combining with equalities (19) and (20)
radic4119899+(119899 + 1) 2119899+1+8119905+[cosh (2)minus3]1198724+[cosh (2)minus10
3]1198725
le 119864119864 (119865119876119899)
(21)
where1198724= 2119885119892(119866)minus 2119898+8119902 119872
5= 30119905 + 10119901+10119903 119901 and
119902 are the numbers of pentagons and quadrangles in 119865119876119899 and
119903 is the number of subgraphs consisting of a triangle with apendent vertex attached
Notice that the equality of (21) holds if and only if theequalities of (19) and (20) hold that is the equality holds ifand only if 120582
1= sdot sdot sdot = 120582
2119899 which is impossible for any 119865119876
119899
with 119899 ge 2 Therefore this implies the results of Theorem 9
We now consider the upper bound for the Estrada indexof 119865119876
119899as follows
Theorem 10 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 2119899minus 1 + 119890
radic(119899+1)2119899
(22)
Proof According to the definition of Estrada index we get
119864119864 (119865119876119899) = 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ sum
119896ge1
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(23)
Notice the inequality
2119899
sum
119894=1
[(120582119894)2
]1198962
le [
2119899
sum
119894=1
(120582119894)2
]
1198962
(24)
substituting inequality (24) into (23) we obtain that
119864119864 (119865119876119899) le 2119899+ sum
119896ge1
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899minus 1 + sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(25)
Since the equality holds in 119865119876119899
2119899
sum
119894=1
(120582119894)2
= (119899 + 1) 2119899 (26)
Hence
119864119864 (119865119876119899) le 2119899minus 1 + sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899minus 1 + sum
119896ge0
radic(119899 + 1)2119899119896
119896
= 2119899minus 1 + 119890
radic(119899+1)2119899
(27)
It is evident that equality of (25) will be attained if andonly if the graph 119865119876
119899has no nonzero eigenvalues which in
turn happens only in the case of the edgeless graph 119870119899 it is
impossible for any 119865119876119899with 119899 ge 2 that directly leads to the
inequality in (27)Hence we can obtain the upper bound for the Estrada
index of 119865119876119899
119864119864 (119865119876119899) lt 2119899minus 1 + 119890
radic(119899+1)2119899
(28)
The proof of Theorem 10 is completed
Remark 11 In [23] it was proved that
119890119903+ (119899 minus 1) 119890
minus119903(119899minus1)le 119864119864 (119866) lt 119899 minus 2 + 119890
119903+ 119890radic119903(119899minus119903)minus1
(29)
with equality holds if and only if 1205822= 1205823= sdot sdot sdot = 120582
119899=
minus119903(119899 minus 1)
Abstract and Applied Analysis 5
Notice that the spectral radius of 119865119876119899is 1205821= 119899 + 1 and
119903 = 119899 +1 applying Lemma6 we also give the lower and upperbounds connecting119864119864(119865119876
119899) and its spectral radius by simple
computations where the equality is impossible for any 119865119876119899
hence
119890119899+1
+ (2119899minus 1) 119890
(minus119899minus1)(2119899minus1)
lt 119864119864 (119865119876119899) lt 2119899minus 2 + 119890
119899+1+ 119890radic(119899+1)[2
119899minus(119899+1)]minus1
(30)
33 Some Properties on Estrada Index Involving Energy of119865119876119899 In this section we investigate the relations between the
Estrada index and the energy of 119865119876119899 We firstly prove the
lower bounds involving energy for the Estrada index of 119865119876119899
inTheorem 12 proof some techniques in [23] are referred to
Theorem 12 For any 119865119876119899with 119899 ge 2 one has
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (31)
Proof Assume that 119899119894denote the number of positive eigen-
values we begin with the definition of Estrada index119864119864(119865119876
119899)
119864119864 (119865119876119899) =
2119899
sum
119894=1
119890120582
119894= sum
120582119894le0
119890120582
119894+ sum
120582119894gt0
119890120582
119894 (32)
Since 119890119909 ge 1 + 119909 with equality holds if and only if 119909 = 0 wehave
sum
120582119894le0
119890120582
119894ge sum
120582119894le0
(1 + 120582119894) = (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
(33)
The other underlying inequality is 119890119909 ge 119890119909 and equalityholds if and only if 119909 = 1 we get
sum
120582119894gt0
119890120582
119894ge sum
120582119894gt0
119890120582119894= 119890 (120582
1+ 1205822sdot sdot sdot + 120582
119899119894) (34)
Substituting the inequalities (33) and (34) into (32)
119864119864 (119865119876119899) ge (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
+ 119890 (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (120582
1+ 1205822sdot sdot sdot + 120582
119899119894+ 120582119899119894+1
+ sdot sdot sdot + 120582119899)
+ (119890 minus 1) (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (119890 minus 1) (120582
1+ 1205822sdot sdot sdot + 120582
119899119894)
(35)
Note that
1205821+ 1205822sdot sdot sdot + 120582
119899119894=1
2119864 (119865119876
119899) (36)
From the above inequalities (35) and (36) we arrive at
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) le 119864119864 (119865119876
119899) (37)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleHence
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (38)
as desired
We now derive the upper bounds involving energy for theEstrada index of 119865119876
119899
Theorem 13 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 119864 (119865119876
119899) + 2119899minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(39)
Proof We consider that
119864119864 (119865119876119899) =
119899
sum
119894=1
119890120582
119894= 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896
le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
(40)
Taking into account the definition of graph energy equa-tion (1) we obtain
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) +
2119899
sum
119894=1
sum
119896ge2
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(41)
In light of the inequality (24) holds for integer 119896 ge 2 weobtain that
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(42)
Substituting (26) into (42) we get
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(43)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleFrom the above argument we get the result ofTheorem 13
6 Abstract and Applied Analysis
4 Conclusions
The main purpose of this paper is to investigate the Estradaindex of 119865119876
119899with 119899 ge 2 we established the explicit formulae
for calculating the Estrada index of 119865119876119899by deducing the
characteristic polynomial of the adjacency matrix in spectralgraph theory
Moreover some lower and upper bounds for Estradaindex of 119865119876
119899were proposed by utilizing the arithmetic and
geometric mean inequality The lower and upper bounds forthe Estrada index involving energy of119865119876
119899were also obtained
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework of Jia Bao Liu was supported by the Natural ScienceFoundation of Anhui Province of China under Grant noKJ2013B105 the work of Xiang-Feng Pan was supported bythe National Science Foundation of China under Grants10901001 11171097 and 11371028
References
[1] J M Xu Topological Strucure and Analysis of InterconnctionNetworks KluwerAcademicDordrechtTheNetherlands 2001
[2] I Gutman ldquoThe energy of a graphrdquo Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz vol 103 no22 pp 100ndash105 1978
[3] E Estrada ldquoCharacterization of 3Dmolecular structurerdquoChem-ical Physics Letters vol 319 no 5 pp 713ndash718 2000
[4] E Estrada and J A Rodrıguez-Velazquez ldquoSubgraph centralityin complex networksrdquo Physical Review E vol 71 no 5 ArticleID 056103 9 pages 2005
[5] E Estrada and J A Rodrıguez-Velazquez ldquoSpectral measures ofbipartivity in complex networksrdquo Physical Review E vol 72 no4 Article ID 046105 6 pages 2005
[6] K C Das I Gutman and B Zhou ldquoNew upper bounds onZagreb indicesrdquo Journal of Mathematical Chemistry vol 46 no2 pp 514ndash521 2009
[7] D M Cvetkovic M Doob I Gutman and A TorgasevRecent Results in the Theory of Graph Spectra vol 36 ofAnnals of Discrete Mathematics North-Holland AmsterdamThe Netherlands 1988
[8] E A Amawy and S Latifi ldquoProperties and performance offolded hypercubesrdquo IEEE Transactions on Parallel and Dis-tributed Systems vol 2 no 1 pp 31ndash42 1991
[9] R Indhumathi and S A Choudum ldquoEmbedding certainheight-balanced trees and complete 119901119898-ary trees into hyper-cubesrdquo Journal of Discrete Algorithms vol 22 pp 53ndash65 2013
[10] J Fink ldquoPerfect matchings extend to Hamilton cycles inhypercubesrdquo Journal of Combinatorial Theory B vol 97 no 6pp 1074ndash1076 2007
[11] J B Liu J D Cao X F Pan and A Elaiw ldquoThe Kirchhoffindex of hypercubes and related complex networksrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 5431897 pages 2013
[12] MChen andBXChen ldquoSpectra of foldedhypercubesrdquo Journalof East China Normal University (Nature Science) vol 2 pp 39ndash46 2011
[13] X He H Liu and Q Liu ldquoCycle embedding in faulty foldedhypercuberdquo International Journal of Applied Mathematics ampStatistics vol 37 no 7 pp 97ndash109 2013
[14] S Cao H Liu and X He ldquoOn constraint fault-free cycles infolded hypercuberdquo International Journal of AppliedMathematicsamp Statistics vol 42 no 12 pp 38ndash44 2013
[15] Y Zhang H Liu and M Liu ldquoCycles embedding on foldedhypercubes with vertex faultsrdquo International Journal of AppliedMathematics amp Statistics vol 41 no 11 pp 58ndash70 2013
[16] X B Chen ldquoConstruction of optimal independent spanningtrees on folded hypercubesrdquo Information Sciences vol 253 pp147ndash156 2013
[17] M Chen X Guo and S Zhai ldquoTotal chromatic number offolded hypercubesrdquo Ars Combinatoria vol 111 pp 265ndash2722013
[18] G H Wen Z S Duan W W Yu and G R Chen ldquoConsensusin multi-agent systems with communication constraintsrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 2pp 170ndash182 2012
[19] J B Liu X F Pan YWang and J D Cao ldquoThe Kirchhoff indexof folded hypercubes and some variant networksrdquoMathematicalProblems in Engineering vol 2014 Article ID 380874 9 pages2014
[20] E Estrada J A Rodrıguez-Velazquez and M Randic ldquoAtomicbranching in moleculesrdquo International Journal of QuantumChemistry vol 106 no 4 pp 823ndash832 2006
[21] B Zhou and Z Du ldquoSome lower bounds for Estrada indexrdquoIranian Journal of Mathematical Chemistry vol 1 no 2 pp 67ndash72 2010
[22] Y Shang ldquoLower bounds for the Estrada index of graphsrdquoElectronic Journal of Linear Algebra vol 23 pp 664ndash668 2012
[23] J P Liu and B L Liu ldquoBounds of the Estrada index of graphsrdquoApplied Mathematics vol 25 no 3 pp 325ndash330 2010
Submit your manuscripts athttpwwwhindawicom
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
119866 The first few spectral moments of an (119899119898)-graph with 119898edges and 119905 triangles satisfy the following relations [4]
1198720=
119899
sum
119894=1
(120582119894)0
= 119899 1198721=
119899
sum
119894=1
(120582119894)1
= 0
1198722=
119899
sum
119894=1
(120582119894)2
= 2119898 1198723=
119899
sum
119894=1
(120582119894)3
= 6119905
(4)
For 1 le 119894 le 119899 let119889119894be the degree of vertex V
119894in119866The first
Zagreb index [6] of the graph119866 is defined as119885119892(119866) = sum119899
119894=11198892
119894
1198724=
119899
sum
119894=1
(120582119894)4
= 2119885119892 (119866) minus 2119898 + 8119902
1198725=
119899
sum
119894=1
(120582119894)5
= 30119905 + 10119901 + 10119903
(5)
where 119901 and 119902 are the numbers of pentagons and quadranglesin119866 and 119903 is the number of subgraphs consisting of a trianglewith a pendent vertex attached [7]
The hypercubes 119876119899is one of the most popular and
efficient interconnection networks due to its many excellentperformances for some practical applications There is alarge amount of literature on the properties of hypercubesnetworks [8ndash11] As an important variant of 119876
119899 the folded
hypercubes networks 119865119876119899 proposed by Amawy and Latifi
[8] are the graphs obtained from 119876119899by adding an edge
between any pair of vertices complementary addresses Thefolded hypercubes 119865119876
119899obtained considerable attention due
to its perfect properties such as symmetry regular structurestrong connectivity small diameter and many of its proper-ties which have been explored [12ndash19]
The remainder of the present paper is organized asfollows In Section 2 we present some basic notations andsome preliminaries in our discussionThe proofs of our mainresults are in Section 3 and some conclusions are given inSection 4 respectively
2 Notations and Some Preliminaries
In this section we introduce some basic properties which willbe used in the proofs of our main results
Let 119875119865119876119899
(119909) be the characteristic polynomial of the adja-cency matrix of the folded hypercube 119865119876
119899 the following
results were shown in [12]
Lemma 1 (see [12]) The characteristic polynomial of theadjacency matrix of the 119865119876
119899(119899 ge 3) is
119875 (119865119876119899 120582) = [120582 minus (119899 minus 7)] [120582 minus (119899 minus 3)]
3119875 (119865119876
119899minus1 120582 minus 1)
times
119899minus2
prod
119894=2
119875 (119865119876119899minus119894 120582 minus (119894 minus 4))
(6)
Lemma 2 (see [12]) For 119865119876119899with 119899 ge 3 the spectrum of
adjacency matrix is as follows(1) If 119899 equiv 0 (mod 2)
119878119901119890119888 (119865119876119899) = (
minus119899 + 1 minus119899 + 5 minus119899 + 9 sdot sdot sdot 119899 minus 7 119899 minus 3 119899 + 1
1198620
119899+ 1198621
1198991198622
119899+ 1198623
1198991198624
119899+ 1198625
119899sdot sdot sdot 119862
119899minus4
119899+ 119862119899minus3
119899119862119899minus2
119899+ 119862119899minus1
119899119862119899
119899
) (7)
(2) if 119899 equiv 1 (mod 2)
119878119901119890119888 (119865119876119899) = (
minus119899 minus 1 minus119899 + 3 minus119899 + 7 sdot sdot sdot 119899 minus 7 119899 minus 3 119899 + 1
1198620
1198991198621
119899+ 1198622
1198991198623
119899+ 1198624
119899sdot sdot sdot 119862
119899minus4
119899+ 119862119899minus3
119899119862119899minus2
119899+ 119862119899minus1
119899119862119899
119899
) (8)
where 119862119894119899are the binomial coefficients and the elements in
the first and second rows are the eigenvalues of the adjacencymatrix of 119865119876
119899and the corresponding multiplicities respec-
tively
Lemma 3 (see [20]) The eigenvalues of a bipartite graphsatisfy the pairing property 120582
119899minus119894+1= 120582119894 119894 = 1 2 119899
Therefore if the graph 119866 is bipartite and if 1205780is nullity (the
multiplicity of its eigenvalue zero) then
119864119864 (119866) = 1205780+ 2sum
+
cosh (119894) (9)
where cosh stands for the hyperbolic cosine cosh(119909) =
(119890119909+ 119890minus119909)2 whereas sum
+denotes summation over all positive
eigenvalues of the corresponding graph
Lemma 4 (see [21]) Let119866 be a graph with119898 edges For 119896 ge 4119872119896+2
ge 119872119896 (10)
with equality for all even 119896 ge 4 if and only if 119866 consists of 119898copies of119870
2and possibly isolated vertices and with equality for
all odd 119896 ge 5 if and only if 119866 is a bipartite graph
The following lemma is an immediate result of theprevious lemma
Abstract and Applied Analysis 3
Lemma 5 (see [22]) Let 119866 be an (119899119898) graph with 119898 edgesFor 119896 ge 4
119899
sum
119894=1
(2120582119894)119896+2
ge 4
119899
sum
119894=1
(2120582119894)119896
(11)
with equality for all even 119896 ge 4 if and only if 119866 consists of 119898copies of119870
2and possibly isolated vertices and with equality for
all odd 119896 ge 5 if and only if 119866 is a bipartite graph
Lemma 6 (see [23]) Let 119866 be a regular graph of degree 119903 = 0
and of order 119899 Then its Estrada index is bounded by
119890119903+ (119899 minus 1) 119890
minus119903(119899minus1)le 119864119864 (119866) lt 119899 minus 2 + 119890
119903+ 119890radic119903(119899minus119903)minus1
(12)
Equality holds if and only if 1205822= 1205823= sdot sdot sdot = 120582
119899= minus119903(119899 minus 1)
Lemma 7 (see [23]) The Estrada index 119864119864(119866) and the graphenergy 119864(119866) satisfy the following inequality
1
2119864 (119866) (119890 minus 1) + 119899 minus 119899
+le 119864119864 (119866) le 119899 minus 1 + 119890
119864(119866)2 (13)
and equalities on both sides hold if and only if 119864(119866) = 0
3 Main Results
31The Estrada Index of FoldedHypercubes Networks119865119876119899 In
this section we present some explicit formulae for calculatingthe Estrada index of 119865119876
119899 For convenience we assume that
119862119894
119899= 0 if 119894 lt 0 or 119894 gt 119899
Theorem 8 For any 119865119876119899with 119899 ge 3 then
(1) 119864119864(119865119876119899) = sum
1198992
119894=0(1198622119894
119899+ 1198622119894+1
119899)1198904119894minus119899+1 119894 = 0 1 1198992
if 119899 equiv 0 (mod 2)
(2) 119864119864(119865119876119899) = sum1198992
119894=0(1198622119894minus1
119899+ 1198622119894
119899)1198904119894minus119899minus1 119894 = 0 1 (119899 +
1)2 if 119899 equiv 1 (mod 2)
where the 4119894 minus 119899 + 1 and 4119894 minus 119899 minus 1 (119894 = 0 1 1198992 or (119899 +1)2) are the eigenvalues of the adjacent matrix of 119865119876
119899and 119862119894
119899
denotes the binomial coefficients
Proof By Lemma 1 the characteristic polynomial of the ad-jacent matrix of 119865119876
119899is
119875 (119865119876119899 120582) = [120582 minus (119899 minus 7)] [120582 minus (119899 minus 3)]
3119875 (119865119876
119899minus1 120582 minus 1)
times
119899minus2
prod
119894=2
119875 (119865119876119899minus119894 120582 minus (119894 minus 4))
(14)
Through calculating eigenvalues of characteristic polyno-mial and its multiplicities we obtained that
(1) if 119899 equiv 0 (mod 2) 119865119876119899have 1198992 + 1 different eigen-
values 4119894 minus 119899 + 1 with the multiplicities 1198622119894119899+ 1198622119894+1
119899
where 119894 = 0 1 1198992
(2) if 119899 equiv 1 (mod 2) 119865119876119899have (119899 + 1)2 different
eigenvalues 4119894 minus 119899 minus 1 with the multiplicities 1198622119894minus1119899
+
1198622119894
119899 where 119894 = 0 1 (119899 + 1)2
Combining with the definition of the Estrada index wederived the result of Theorem 8
32 Some Bounds for the Estrada Index of Folded HypercubesNetworks 119865119876
119899 It is well known that 119865119876
119899have 2119899 vertices
Let 1205821ge 1205822ge sdot sdot sdot ge 120582
119899ge 120582119899+1
ge sdot sdot sdot ge 1205822119899 be the
eigenvalues of 119865119876119899with nonincreasing order In order to
obtain the bounds for the Estrada index of 119865119876119899 we prove
some results by utilizing the arithmetic and geometric meaninequality in our proof some techniques in [22] are referredto
Theorem 9 For any 119865119876119899with 119899 ge 2 one has
radic4119899+(119899 + 1) 2119899+1+8119905+[cosh (2)minus3]1198724+[cosh (2)minus10
3]1198725
lt 119864119864 (119865119876119899)
(15)
where 1198724= 2119885119892(119866) minus 2119898 + 8119902 119872
5= 30119905 + 10119901 + 10119903 119901
and 119902 are the numbers of pentagons and quadrangles in 119865119876119899
and 119903 is the number of subgraphs consisting of a triangle witha pendent vertex attached
Proof In order to obtain the lower bounds for the Estradaindex consider that
1198641198642(119865119876119899) =
2119899
sum
119894=1
1198902120582119894 + 2sum
119894lt119895
119890120582119894119890120582119895 (16)
Noting that 1198720= 2119899 1198721= 0 119872
2= (119899 + 1)2
119899minus1 and1198723= 6119905 we obtain
2119899
sum
119894=1
1198902120582119894 =
2119899
sum
119894=1
sum
119896ge0
(2120582119894)119896
119896
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 +
2119899
sum
119894=1
sum
119896ge4
(2120582119894)119896
119896
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 + sum
119896ge2
1
(2119896)
2119899
sum
119894=1
(2120582119894)2119896
+ sum
119896ge2
1
(2119896 + 1)
2119899
sum
119894=1
(2120582119894)2119896+1
(17)
By Lemma 5
119899
sum
119894=1
(2120582119894)119896+2
ge 4
119899
sum
119894=1
(2120582119894)119896
(18)
4 Abstract and Applied Analysis
we can get that
2119899
sum
119894=1
1198902120582119894 ge 2
119899+ (119899 + 1) 2
119899+1+ 8119905
+ sum
119896ge2
1
(2119896)
2119899
sum
119894=1
22119896minus4
(2120582119894)4
+ sum
119896ge2
1
(2119896 + 1)
2119899
sum
119894=1
22119896minus4
(2120582119894)5
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 + [cosh (2) minus 3]1198724
+ [cosh (2) minus 10
3]1198725
(19)
where1198724= 2119885119892(119866)minus 2119898+8119902 119872
5= 30119905 + 10119901+10119903 119901 and
119902 are the numbers of pentagons and quadrangles in 119865119876119899 and
119903 is the number of subgraphs consisting of a triangle with apendent vertex attached
As for the terms 2sum119894lt119895
119890120582119894119890120582119895 by the arithmetic and
geometric mean inequality and the fact that1198721= 0
2sum
119894lt119895
119890120582119894119890120582119895 ge 2
119899(2119899minus 1)(prod
119894lt119895
119890120582119894119890120582119895)
22119899(2119899minus1)
= 2119899(2119899minus 1)[
[
(prod
119894=1
119890120582119894)
2119899minus1
]
]
22119899(2119899minus1)
= 2119899(2119899minus 1) (119890
1198721)22119899
= 2119899(2119899minus 1)
(20)
where the equality holds if and only if 1205821= sdot sdot sdot = 120582
2119899
Combining with equalities (19) and (20)
radic4119899+(119899 + 1) 2119899+1+8119905+[cosh (2)minus3]1198724+[cosh (2)minus10
3]1198725
le 119864119864 (119865119876119899)
(21)
where1198724= 2119885119892(119866)minus 2119898+8119902 119872
5= 30119905 + 10119901+10119903 119901 and
119902 are the numbers of pentagons and quadrangles in 119865119876119899 and
119903 is the number of subgraphs consisting of a triangle with apendent vertex attached
Notice that the equality of (21) holds if and only if theequalities of (19) and (20) hold that is the equality holds ifand only if 120582
1= sdot sdot sdot = 120582
2119899 which is impossible for any 119865119876
119899
with 119899 ge 2 Therefore this implies the results of Theorem 9
We now consider the upper bound for the Estrada indexof 119865119876
119899as follows
Theorem 10 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 2119899minus 1 + 119890
radic(119899+1)2119899
(22)
Proof According to the definition of Estrada index we get
119864119864 (119865119876119899) = 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ sum
119896ge1
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(23)
Notice the inequality
2119899
sum
119894=1
[(120582119894)2
]1198962
le [
2119899
sum
119894=1
(120582119894)2
]
1198962
(24)
substituting inequality (24) into (23) we obtain that
119864119864 (119865119876119899) le 2119899+ sum
119896ge1
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899minus 1 + sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(25)
Since the equality holds in 119865119876119899
2119899
sum
119894=1
(120582119894)2
= (119899 + 1) 2119899 (26)
Hence
119864119864 (119865119876119899) le 2119899minus 1 + sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899minus 1 + sum
119896ge0
radic(119899 + 1)2119899119896
119896
= 2119899minus 1 + 119890
radic(119899+1)2119899
(27)
It is evident that equality of (25) will be attained if andonly if the graph 119865119876
119899has no nonzero eigenvalues which in
turn happens only in the case of the edgeless graph 119870119899 it is
impossible for any 119865119876119899with 119899 ge 2 that directly leads to the
inequality in (27)Hence we can obtain the upper bound for the Estrada
index of 119865119876119899
119864119864 (119865119876119899) lt 2119899minus 1 + 119890
radic(119899+1)2119899
(28)
The proof of Theorem 10 is completed
Remark 11 In [23] it was proved that
119890119903+ (119899 minus 1) 119890
minus119903(119899minus1)le 119864119864 (119866) lt 119899 minus 2 + 119890
119903+ 119890radic119903(119899minus119903)minus1
(29)
with equality holds if and only if 1205822= 1205823= sdot sdot sdot = 120582
119899=
minus119903(119899 minus 1)
Abstract and Applied Analysis 5
Notice that the spectral radius of 119865119876119899is 1205821= 119899 + 1 and
119903 = 119899 +1 applying Lemma6 we also give the lower and upperbounds connecting119864119864(119865119876
119899) and its spectral radius by simple
computations where the equality is impossible for any 119865119876119899
hence
119890119899+1
+ (2119899minus 1) 119890
(minus119899minus1)(2119899minus1)
lt 119864119864 (119865119876119899) lt 2119899minus 2 + 119890
119899+1+ 119890radic(119899+1)[2
119899minus(119899+1)]minus1
(30)
33 Some Properties on Estrada Index Involving Energy of119865119876119899 In this section we investigate the relations between the
Estrada index and the energy of 119865119876119899 We firstly prove the
lower bounds involving energy for the Estrada index of 119865119876119899
inTheorem 12 proof some techniques in [23] are referred to
Theorem 12 For any 119865119876119899with 119899 ge 2 one has
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (31)
Proof Assume that 119899119894denote the number of positive eigen-
values we begin with the definition of Estrada index119864119864(119865119876
119899)
119864119864 (119865119876119899) =
2119899
sum
119894=1
119890120582
119894= sum
120582119894le0
119890120582
119894+ sum
120582119894gt0
119890120582
119894 (32)
Since 119890119909 ge 1 + 119909 with equality holds if and only if 119909 = 0 wehave
sum
120582119894le0
119890120582
119894ge sum
120582119894le0
(1 + 120582119894) = (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
(33)
The other underlying inequality is 119890119909 ge 119890119909 and equalityholds if and only if 119909 = 1 we get
sum
120582119894gt0
119890120582
119894ge sum
120582119894gt0
119890120582119894= 119890 (120582
1+ 1205822sdot sdot sdot + 120582
119899119894) (34)
Substituting the inequalities (33) and (34) into (32)
119864119864 (119865119876119899) ge (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
+ 119890 (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (120582
1+ 1205822sdot sdot sdot + 120582
119899119894+ 120582119899119894+1
+ sdot sdot sdot + 120582119899)
+ (119890 minus 1) (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (119890 minus 1) (120582
1+ 1205822sdot sdot sdot + 120582
119899119894)
(35)
Note that
1205821+ 1205822sdot sdot sdot + 120582
119899119894=1
2119864 (119865119876
119899) (36)
From the above inequalities (35) and (36) we arrive at
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) le 119864119864 (119865119876
119899) (37)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleHence
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (38)
as desired
We now derive the upper bounds involving energy for theEstrada index of 119865119876
119899
Theorem 13 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 119864 (119865119876
119899) + 2119899minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(39)
Proof We consider that
119864119864 (119865119876119899) =
119899
sum
119894=1
119890120582
119894= 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896
le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
(40)
Taking into account the definition of graph energy equa-tion (1) we obtain
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) +
2119899
sum
119894=1
sum
119896ge2
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(41)
In light of the inequality (24) holds for integer 119896 ge 2 weobtain that
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(42)
Substituting (26) into (42) we get
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(43)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleFrom the above argument we get the result ofTheorem 13
6 Abstract and Applied Analysis
4 Conclusions
The main purpose of this paper is to investigate the Estradaindex of 119865119876
119899with 119899 ge 2 we established the explicit formulae
for calculating the Estrada index of 119865119876119899by deducing the
characteristic polynomial of the adjacency matrix in spectralgraph theory
Moreover some lower and upper bounds for Estradaindex of 119865119876
119899were proposed by utilizing the arithmetic and
geometric mean inequality The lower and upper bounds forthe Estrada index involving energy of119865119876
119899were also obtained
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework of Jia Bao Liu was supported by the Natural ScienceFoundation of Anhui Province of China under Grant noKJ2013B105 the work of Xiang-Feng Pan was supported bythe National Science Foundation of China under Grants10901001 11171097 and 11371028
References
[1] J M Xu Topological Strucure and Analysis of InterconnctionNetworks KluwerAcademicDordrechtTheNetherlands 2001
[2] I Gutman ldquoThe energy of a graphrdquo Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz vol 103 no22 pp 100ndash105 1978
[3] E Estrada ldquoCharacterization of 3Dmolecular structurerdquoChem-ical Physics Letters vol 319 no 5 pp 713ndash718 2000
[4] E Estrada and J A Rodrıguez-Velazquez ldquoSubgraph centralityin complex networksrdquo Physical Review E vol 71 no 5 ArticleID 056103 9 pages 2005
[5] E Estrada and J A Rodrıguez-Velazquez ldquoSpectral measures ofbipartivity in complex networksrdquo Physical Review E vol 72 no4 Article ID 046105 6 pages 2005
[6] K C Das I Gutman and B Zhou ldquoNew upper bounds onZagreb indicesrdquo Journal of Mathematical Chemistry vol 46 no2 pp 514ndash521 2009
[7] D M Cvetkovic M Doob I Gutman and A TorgasevRecent Results in the Theory of Graph Spectra vol 36 ofAnnals of Discrete Mathematics North-Holland AmsterdamThe Netherlands 1988
[8] E A Amawy and S Latifi ldquoProperties and performance offolded hypercubesrdquo IEEE Transactions on Parallel and Dis-tributed Systems vol 2 no 1 pp 31ndash42 1991
[9] R Indhumathi and S A Choudum ldquoEmbedding certainheight-balanced trees and complete 119901119898-ary trees into hyper-cubesrdquo Journal of Discrete Algorithms vol 22 pp 53ndash65 2013
[10] J Fink ldquoPerfect matchings extend to Hamilton cycles inhypercubesrdquo Journal of Combinatorial Theory B vol 97 no 6pp 1074ndash1076 2007
[11] J B Liu J D Cao X F Pan and A Elaiw ldquoThe Kirchhoffindex of hypercubes and related complex networksrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 5431897 pages 2013
[12] MChen andBXChen ldquoSpectra of foldedhypercubesrdquo Journalof East China Normal University (Nature Science) vol 2 pp 39ndash46 2011
[13] X He H Liu and Q Liu ldquoCycle embedding in faulty foldedhypercuberdquo International Journal of Applied Mathematics ampStatistics vol 37 no 7 pp 97ndash109 2013
[14] S Cao H Liu and X He ldquoOn constraint fault-free cycles infolded hypercuberdquo International Journal of AppliedMathematicsamp Statistics vol 42 no 12 pp 38ndash44 2013
[15] Y Zhang H Liu and M Liu ldquoCycles embedding on foldedhypercubes with vertex faultsrdquo International Journal of AppliedMathematics amp Statistics vol 41 no 11 pp 58ndash70 2013
[16] X B Chen ldquoConstruction of optimal independent spanningtrees on folded hypercubesrdquo Information Sciences vol 253 pp147ndash156 2013
[17] M Chen X Guo and S Zhai ldquoTotal chromatic number offolded hypercubesrdquo Ars Combinatoria vol 111 pp 265ndash2722013
[18] G H Wen Z S Duan W W Yu and G R Chen ldquoConsensusin multi-agent systems with communication constraintsrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 2pp 170ndash182 2012
[19] J B Liu X F Pan YWang and J D Cao ldquoThe Kirchhoff indexof folded hypercubes and some variant networksrdquoMathematicalProblems in Engineering vol 2014 Article ID 380874 9 pages2014
[20] E Estrada J A Rodrıguez-Velazquez and M Randic ldquoAtomicbranching in moleculesrdquo International Journal of QuantumChemistry vol 106 no 4 pp 823ndash832 2006
[21] B Zhou and Z Du ldquoSome lower bounds for Estrada indexrdquoIranian Journal of Mathematical Chemistry vol 1 no 2 pp 67ndash72 2010
[22] Y Shang ldquoLower bounds for the Estrada index of graphsrdquoElectronic Journal of Linear Algebra vol 23 pp 664ndash668 2012
[23] J P Liu and B L Liu ldquoBounds of the Estrada index of graphsrdquoApplied Mathematics vol 25 no 3 pp 325ndash330 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Lemma 5 (see [22]) Let 119866 be an (119899119898) graph with 119898 edgesFor 119896 ge 4
119899
sum
119894=1
(2120582119894)119896+2
ge 4
119899
sum
119894=1
(2120582119894)119896
(11)
with equality for all even 119896 ge 4 if and only if 119866 consists of 119898copies of119870
2and possibly isolated vertices and with equality for
all odd 119896 ge 5 if and only if 119866 is a bipartite graph
Lemma 6 (see [23]) Let 119866 be a regular graph of degree 119903 = 0
and of order 119899 Then its Estrada index is bounded by
119890119903+ (119899 minus 1) 119890
minus119903(119899minus1)le 119864119864 (119866) lt 119899 minus 2 + 119890
119903+ 119890radic119903(119899minus119903)minus1
(12)
Equality holds if and only if 1205822= 1205823= sdot sdot sdot = 120582
119899= minus119903(119899 minus 1)
Lemma 7 (see [23]) The Estrada index 119864119864(119866) and the graphenergy 119864(119866) satisfy the following inequality
1
2119864 (119866) (119890 minus 1) + 119899 minus 119899
+le 119864119864 (119866) le 119899 minus 1 + 119890
119864(119866)2 (13)
and equalities on both sides hold if and only if 119864(119866) = 0
3 Main Results
31The Estrada Index of FoldedHypercubes Networks119865119876119899 In
this section we present some explicit formulae for calculatingthe Estrada index of 119865119876
119899 For convenience we assume that
119862119894
119899= 0 if 119894 lt 0 or 119894 gt 119899
Theorem 8 For any 119865119876119899with 119899 ge 3 then
(1) 119864119864(119865119876119899) = sum
1198992
119894=0(1198622119894
119899+ 1198622119894+1
119899)1198904119894minus119899+1 119894 = 0 1 1198992
if 119899 equiv 0 (mod 2)
(2) 119864119864(119865119876119899) = sum1198992
119894=0(1198622119894minus1
119899+ 1198622119894
119899)1198904119894minus119899minus1 119894 = 0 1 (119899 +
1)2 if 119899 equiv 1 (mod 2)
where the 4119894 minus 119899 + 1 and 4119894 minus 119899 minus 1 (119894 = 0 1 1198992 or (119899 +1)2) are the eigenvalues of the adjacent matrix of 119865119876
119899and 119862119894
119899
denotes the binomial coefficients
Proof By Lemma 1 the characteristic polynomial of the ad-jacent matrix of 119865119876
119899is
119875 (119865119876119899 120582) = [120582 minus (119899 minus 7)] [120582 minus (119899 minus 3)]
3119875 (119865119876
119899minus1 120582 minus 1)
times
119899minus2
prod
119894=2
119875 (119865119876119899minus119894 120582 minus (119894 minus 4))
(14)
Through calculating eigenvalues of characteristic polyno-mial and its multiplicities we obtained that
(1) if 119899 equiv 0 (mod 2) 119865119876119899have 1198992 + 1 different eigen-
values 4119894 minus 119899 + 1 with the multiplicities 1198622119894119899+ 1198622119894+1
119899
where 119894 = 0 1 1198992
(2) if 119899 equiv 1 (mod 2) 119865119876119899have (119899 + 1)2 different
eigenvalues 4119894 minus 119899 minus 1 with the multiplicities 1198622119894minus1119899
+
1198622119894
119899 where 119894 = 0 1 (119899 + 1)2
Combining with the definition of the Estrada index wederived the result of Theorem 8
32 Some Bounds for the Estrada Index of Folded HypercubesNetworks 119865119876
119899 It is well known that 119865119876
119899have 2119899 vertices
Let 1205821ge 1205822ge sdot sdot sdot ge 120582
119899ge 120582119899+1
ge sdot sdot sdot ge 1205822119899 be the
eigenvalues of 119865119876119899with nonincreasing order In order to
obtain the bounds for the Estrada index of 119865119876119899 we prove
some results by utilizing the arithmetic and geometric meaninequality in our proof some techniques in [22] are referredto
Theorem 9 For any 119865119876119899with 119899 ge 2 one has
radic4119899+(119899 + 1) 2119899+1+8119905+[cosh (2)minus3]1198724+[cosh (2)minus10
3]1198725
lt 119864119864 (119865119876119899)
(15)
where 1198724= 2119885119892(119866) minus 2119898 + 8119902 119872
5= 30119905 + 10119901 + 10119903 119901
and 119902 are the numbers of pentagons and quadrangles in 119865119876119899
and 119903 is the number of subgraphs consisting of a triangle witha pendent vertex attached
Proof In order to obtain the lower bounds for the Estradaindex consider that
1198641198642(119865119876119899) =
2119899
sum
119894=1
1198902120582119894 + 2sum
119894lt119895
119890120582119894119890120582119895 (16)
Noting that 1198720= 2119899 1198721= 0 119872
2= (119899 + 1)2
119899minus1 and1198723= 6119905 we obtain
2119899
sum
119894=1
1198902120582119894 =
2119899
sum
119894=1
sum
119896ge0
(2120582119894)119896
119896
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 +
2119899
sum
119894=1
sum
119896ge4
(2120582119894)119896
119896
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 + sum
119896ge2
1
(2119896)
2119899
sum
119894=1
(2120582119894)2119896
+ sum
119896ge2
1
(2119896 + 1)
2119899
sum
119894=1
(2120582119894)2119896+1
(17)
By Lemma 5
119899
sum
119894=1
(2120582119894)119896+2
ge 4
119899
sum
119894=1
(2120582119894)119896
(18)
4 Abstract and Applied Analysis
we can get that
2119899
sum
119894=1
1198902120582119894 ge 2
119899+ (119899 + 1) 2
119899+1+ 8119905
+ sum
119896ge2
1
(2119896)
2119899
sum
119894=1
22119896minus4
(2120582119894)4
+ sum
119896ge2
1
(2119896 + 1)
2119899
sum
119894=1
22119896minus4
(2120582119894)5
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 + [cosh (2) minus 3]1198724
+ [cosh (2) minus 10
3]1198725
(19)
where1198724= 2119885119892(119866)minus 2119898+8119902 119872
5= 30119905 + 10119901+10119903 119901 and
119902 are the numbers of pentagons and quadrangles in 119865119876119899 and
119903 is the number of subgraphs consisting of a triangle with apendent vertex attached
As for the terms 2sum119894lt119895
119890120582119894119890120582119895 by the arithmetic and
geometric mean inequality and the fact that1198721= 0
2sum
119894lt119895
119890120582119894119890120582119895 ge 2
119899(2119899minus 1)(prod
119894lt119895
119890120582119894119890120582119895)
22119899(2119899minus1)
= 2119899(2119899minus 1)[
[
(prod
119894=1
119890120582119894)
2119899minus1
]
]
22119899(2119899minus1)
= 2119899(2119899minus 1) (119890
1198721)22119899
= 2119899(2119899minus 1)
(20)
where the equality holds if and only if 1205821= sdot sdot sdot = 120582
2119899
Combining with equalities (19) and (20)
radic4119899+(119899 + 1) 2119899+1+8119905+[cosh (2)minus3]1198724+[cosh (2)minus10
3]1198725
le 119864119864 (119865119876119899)
(21)
where1198724= 2119885119892(119866)minus 2119898+8119902 119872
5= 30119905 + 10119901+10119903 119901 and
119902 are the numbers of pentagons and quadrangles in 119865119876119899 and
119903 is the number of subgraphs consisting of a triangle with apendent vertex attached
Notice that the equality of (21) holds if and only if theequalities of (19) and (20) hold that is the equality holds ifand only if 120582
1= sdot sdot sdot = 120582
2119899 which is impossible for any 119865119876
119899
with 119899 ge 2 Therefore this implies the results of Theorem 9
We now consider the upper bound for the Estrada indexof 119865119876
119899as follows
Theorem 10 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 2119899minus 1 + 119890
radic(119899+1)2119899
(22)
Proof According to the definition of Estrada index we get
119864119864 (119865119876119899) = 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ sum
119896ge1
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(23)
Notice the inequality
2119899
sum
119894=1
[(120582119894)2
]1198962
le [
2119899
sum
119894=1
(120582119894)2
]
1198962
(24)
substituting inequality (24) into (23) we obtain that
119864119864 (119865119876119899) le 2119899+ sum
119896ge1
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899minus 1 + sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(25)
Since the equality holds in 119865119876119899
2119899
sum
119894=1
(120582119894)2
= (119899 + 1) 2119899 (26)
Hence
119864119864 (119865119876119899) le 2119899minus 1 + sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899minus 1 + sum
119896ge0
radic(119899 + 1)2119899119896
119896
= 2119899minus 1 + 119890
radic(119899+1)2119899
(27)
It is evident that equality of (25) will be attained if andonly if the graph 119865119876
119899has no nonzero eigenvalues which in
turn happens only in the case of the edgeless graph 119870119899 it is
impossible for any 119865119876119899with 119899 ge 2 that directly leads to the
inequality in (27)Hence we can obtain the upper bound for the Estrada
index of 119865119876119899
119864119864 (119865119876119899) lt 2119899minus 1 + 119890
radic(119899+1)2119899
(28)
The proof of Theorem 10 is completed
Remark 11 In [23] it was proved that
119890119903+ (119899 minus 1) 119890
minus119903(119899minus1)le 119864119864 (119866) lt 119899 minus 2 + 119890
119903+ 119890radic119903(119899minus119903)minus1
(29)
with equality holds if and only if 1205822= 1205823= sdot sdot sdot = 120582
119899=
minus119903(119899 minus 1)
Abstract and Applied Analysis 5
Notice that the spectral radius of 119865119876119899is 1205821= 119899 + 1 and
119903 = 119899 +1 applying Lemma6 we also give the lower and upperbounds connecting119864119864(119865119876
119899) and its spectral radius by simple
computations where the equality is impossible for any 119865119876119899
hence
119890119899+1
+ (2119899minus 1) 119890
(minus119899minus1)(2119899minus1)
lt 119864119864 (119865119876119899) lt 2119899minus 2 + 119890
119899+1+ 119890radic(119899+1)[2
119899minus(119899+1)]minus1
(30)
33 Some Properties on Estrada Index Involving Energy of119865119876119899 In this section we investigate the relations between the
Estrada index and the energy of 119865119876119899 We firstly prove the
lower bounds involving energy for the Estrada index of 119865119876119899
inTheorem 12 proof some techniques in [23] are referred to
Theorem 12 For any 119865119876119899with 119899 ge 2 one has
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (31)
Proof Assume that 119899119894denote the number of positive eigen-
values we begin with the definition of Estrada index119864119864(119865119876
119899)
119864119864 (119865119876119899) =
2119899
sum
119894=1
119890120582
119894= sum
120582119894le0
119890120582
119894+ sum
120582119894gt0
119890120582
119894 (32)
Since 119890119909 ge 1 + 119909 with equality holds if and only if 119909 = 0 wehave
sum
120582119894le0
119890120582
119894ge sum
120582119894le0
(1 + 120582119894) = (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
(33)
The other underlying inequality is 119890119909 ge 119890119909 and equalityholds if and only if 119909 = 1 we get
sum
120582119894gt0
119890120582
119894ge sum
120582119894gt0
119890120582119894= 119890 (120582
1+ 1205822sdot sdot sdot + 120582
119899119894) (34)
Substituting the inequalities (33) and (34) into (32)
119864119864 (119865119876119899) ge (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
+ 119890 (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (120582
1+ 1205822sdot sdot sdot + 120582
119899119894+ 120582119899119894+1
+ sdot sdot sdot + 120582119899)
+ (119890 minus 1) (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (119890 minus 1) (120582
1+ 1205822sdot sdot sdot + 120582
119899119894)
(35)
Note that
1205821+ 1205822sdot sdot sdot + 120582
119899119894=1
2119864 (119865119876
119899) (36)
From the above inequalities (35) and (36) we arrive at
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) le 119864119864 (119865119876
119899) (37)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleHence
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (38)
as desired
We now derive the upper bounds involving energy for theEstrada index of 119865119876
119899
Theorem 13 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 119864 (119865119876
119899) + 2119899minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(39)
Proof We consider that
119864119864 (119865119876119899) =
119899
sum
119894=1
119890120582
119894= 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896
le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
(40)
Taking into account the definition of graph energy equa-tion (1) we obtain
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) +
2119899
sum
119894=1
sum
119896ge2
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(41)
In light of the inequality (24) holds for integer 119896 ge 2 weobtain that
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(42)
Substituting (26) into (42) we get
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(43)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleFrom the above argument we get the result ofTheorem 13
6 Abstract and Applied Analysis
4 Conclusions
The main purpose of this paper is to investigate the Estradaindex of 119865119876
119899with 119899 ge 2 we established the explicit formulae
for calculating the Estrada index of 119865119876119899by deducing the
characteristic polynomial of the adjacency matrix in spectralgraph theory
Moreover some lower and upper bounds for Estradaindex of 119865119876
119899were proposed by utilizing the arithmetic and
geometric mean inequality The lower and upper bounds forthe Estrada index involving energy of119865119876
119899were also obtained
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework of Jia Bao Liu was supported by the Natural ScienceFoundation of Anhui Province of China under Grant noKJ2013B105 the work of Xiang-Feng Pan was supported bythe National Science Foundation of China under Grants10901001 11171097 and 11371028
References
[1] J M Xu Topological Strucure and Analysis of InterconnctionNetworks KluwerAcademicDordrechtTheNetherlands 2001
[2] I Gutman ldquoThe energy of a graphrdquo Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz vol 103 no22 pp 100ndash105 1978
[3] E Estrada ldquoCharacterization of 3Dmolecular structurerdquoChem-ical Physics Letters vol 319 no 5 pp 713ndash718 2000
[4] E Estrada and J A Rodrıguez-Velazquez ldquoSubgraph centralityin complex networksrdquo Physical Review E vol 71 no 5 ArticleID 056103 9 pages 2005
[5] E Estrada and J A Rodrıguez-Velazquez ldquoSpectral measures ofbipartivity in complex networksrdquo Physical Review E vol 72 no4 Article ID 046105 6 pages 2005
[6] K C Das I Gutman and B Zhou ldquoNew upper bounds onZagreb indicesrdquo Journal of Mathematical Chemistry vol 46 no2 pp 514ndash521 2009
[7] D M Cvetkovic M Doob I Gutman and A TorgasevRecent Results in the Theory of Graph Spectra vol 36 ofAnnals of Discrete Mathematics North-Holland AmsterdamThe Netherlands 1988
[8] E A Amawy and S Latifi ldquoProperties and performance offolded hypercubesrdquo IEEE Transactions on Parallel and Dis-tributed Systems vol 2 no 1 pp 31ndash42 1991
[9] R Indhumathi and S A Choudum ldquoEmbedding certainheight-balanced trees and complete 119901119898-ary trees into hyper-cubesrdquo Journal of Discrete Algorithms vol 22 pp 53ndash65 2013
[10] J Fink ldquoPerfect matchings extend to Hamilton cycles inhypercubesrdquo Journal of Combinatorial Theory B vol 97 no 6pp 1074ndash1076 2007
[11] J B Liu J D Cao X F Pan and A Elaiw ldquoThe Kirchhoffindex of hypercubes and related complex networksrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 5431897 pages 2013
[12] MChen andBXChen ldquoSpectra of foldedhypercubesrdquo Journalof East China Normal University (Nature Science) vol 2 pp 39ndash46 2011
[13] X He H Liu and Q Liu ldquoCycle embedding in faulty foldedhypercuberdquo International Journal of Applied Mathematics ampStatistics vol 37 no 7 pp 97ndash109 2013
[14] S Cao H Liu and X He ldquoOn constraint fault-free cycles infolded hypercuberdquo International Journal of AppliedMathematicsamp Statistics vol 42 no 12 pp 38ndash44 2013
[15] Y Zhang H Liu and M Liu ldquoCycles embedding on foldedhypercubes with vertex faultsrdquo International Journal of AppliedMathematics amp Statistics vol 41 no 11 pp 58ndash70 2013
[16] X B Chen ldquoConstruction of optimal independent spanningtrees on folded hypercubesrdquo Information Sciences vol 253 pp147ndash156 2013
[17] M Chen X Guo and S Zhai ldquoTotal chromatic number offolded hypercubesrdquo Ars Combinatoria vol 111 pp 265ndash2722013
[18] G H Wen Z S Duan W W Yu and G R Chen ldquoConsensusin multi-agent systems with communication constraintsrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 2pp 170ndash182 2012
[19] J B Liu X F Pan YWang and J D Cao ldquoThe Kirchhoff indexof folded hypercubes and some variant networksrdquoMathematicalProblems in Engineering vol 2014 Article ID 380874 9 pages2014
[20] E Estrada J A Rodrıguez-Velazquez and M Randic ldquoAtomicbranching in moleculesrdquo International Journal of QuantumChemistry vol 106 no 4 pp 823ndash832 2006
[21] B Zhou and Z Du ldquoSome lower bounds for Estrada indexrdquoIranian Journal of Mathematical Chemistry vol 1 no 2 pp 67ndash72 2010
[22] Y Shang ldquoLower bounds for the Estrada index of graphsrdquoElectronic Journal of Linear Algebra vol 23 pp 664ndash668 2012
[23] J P Liu and B L Liu ldquoBounds of the Estrada index of graphsrdquoApplied Mathematics vol 25 no 3 pp 325ndash330 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
we can get that
2119899
sum
119894=1
1198902120582119894 ge 2
119899+ (119899 + 1) 2
119899+1+ 8119905
+ sum
119896ge2
1
(2119896)
2119899
sum
119894=1
22119896minus4
(2120582119894)4
+ sum
119896ge2
1
(2119896 + 1)
2119899
sum
119894=1
22119896minus4
(2120582119894)5
= 2119899+ (119899 + 1) 2
119899+1+ 8119905 + [cosh (2) minus 3]1198724
+ [cosh (2) minus 10
3]1198725
(19)
where1198724= 2119885119892(119866)minus 2119898+8119902 119872
5= 30119905 + 10119901+10119903 119901 and
119902 are the numbers of pentagons and quadrangles in 119865119876119899 and
119903 is the number of subgraphs consisting of a triangle with apendent vertex attached
As for the terms 2sum119894lt119895
119890120582119894119890120582119895 by the arithmetic and
geometric mean inequality and the fact that1198721= 0
2sum
119894lt119895
119890120582119894119890120582119895 ge 2
119899(2119899minus 1)(prod
119894lt119895
119890120582119894119890120582119895)
22119899(2119899minus1)
= 2119899(2119899minus 1)[
[
(prod
119894=1
119890120582119894)
2119899minus1
]
]
22119899(2119899minus1)
= 2119899(2119899minus 1) (119890
1198721)22119899
= 2119899(2119899minus 1)
(20)
where the equality holds if and only if 1205821= sdot sdot sdot = 120582
2119899
Combining with equalities (19) and (20)
radic4119899+(119899 + 1) 2119899+1+8119905+[cosh (2)minus3]1198724+[cosh (2)minus10
3]1198725
le 119864119864 (119865119876119899)
(21)
where1198724= 2119885119892(119866)minus 2119898+8119902 119872
5= 30119905 + 10119901+10119903 119901 and
119902 are the numbers of pentagons and quadrangles in 119865119876119899 and
119903 is the number of subgraphs consisting of a triangle with apendent vertex attached
Notice that the equality of (21) holds if and only if theequalities of (19) and (20) hold that is the equality holds ifand only if 120582
1= sdot sdot sdot = 120582
2119899 which is impossible for any 119865119876
119899
with 119899 ge 2 Therefore this implies the results of Theorem 9
We now consider the upper bound for the Estrada indexof 119865119876
119899as follows
Theorem 10 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 2119899minus 1 + 119890
radic(119899+1)2119899
(22)
Proof According to the definition of Estrada index we get
119864119864 (119865119876119899) = 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ sum
119896ge1
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(23)
Notice the inequality
2119899
sum
119894=1
[(120582119894)2
]1198962
le [
2119899
sum
119894=1
(120582119894)2
]
1198962
(24)
substituting inequality (24) into (23) we obtain that
119864119864 (119865119876119899) le 2119899+ sum
119896ge1
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899minus 1 + sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(25)
Since the equality holds in 119865119876119899
2119899
sum
119894=1
(120582119894)2
= (119899 + 1) 2119899 (26)
Hence
119864119864 (119865119876119899) le 2119899minus 1 + sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899minus 1 + sum
119896ge0
radic(119899 + 1)2119899119896
119896
= 2119899minus 1 + 119890
radic(119899+1)2119899
(27)
It is evident that equality of (25) will be attained if andonly if the graph 119865119876
119899has no nonzero eigenvalues which in
turn happens only in the case of the edgeless graph 119870119899 it is
impossible for any 119865119876119899with 119899 ge 2 that directly leads to the
inequality in (27)Hence we can obtain the upper bound for the Estrada
index of 119865119876119899
119864119864 (119865119876119899) lt 2119899minus 1 + 119890
radic(119899+1)2119899
(28)
The proof of Theorem 10 is completed
Remark 11 In [23] it was proved that
119890119903+ (119899 minus 1) 119890
minus119903(119899minus1)le 119864119864 (119866) lt 119899 minus 2 + 119890
119903+ 119890radic119903(119899minus119903)minus1
(29)
with equality holds if and only if 1205822= 1205823= sdot sdot sdot = 120582
119899=
minus119903(119899 minus 1)
Abstract and Applied Analysis 5
Notice that the spectral radius of 119865119876119899is 1205821= 119899 + 1 and
119903 = 119899 +1 applying Lemma6 we also give the lower and upperbounds connecting119864119864(119865119876
119899) and its spectral radius by simple
computations where the equality is impossible for any 119865119876119899
hence
119890119899+1
+ (2119899minus 1) 119890
(minus119899minus1)(2119899minus1)
lt 119864119864 (119865119876119899) lt 2119899minus 2 + 119890
119899+1+ 119890radic(119899+1)[2
119899minus(119899+1)]minus1
(30)
33 Some Properties on Estrada Index Involving Energy of119865119876119899 In this section we investigate the relations between the
Estrada index and the energy of 119865119876119899 We firstly prove the
lower bounds involving energy for the Estrada index of 119865119876119899
inTheorem 12 proof some techniques in [23] are referred to
Theorem 12 For any 119865119876119899with 119899 ge 2 one has
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (31)
Proof Assume that 119899119894denote the number of positive eigen-
values we begin with the definition of Estrada index119864119864(119865119876
119899)
119864119864 (119865119876119899) =
2119899
sum
119894=1
119890120582
119894= sum
120582119894le0
119890120582
119894+ sum
120582119894gt0
119890120582
119894 (32)
Since 119890119909 ge 1 + 119909 with equality holds if and only if 119909 = 0 wehave
sum
120582119894le0
119890120582
119894ge sum
120582119894le0
(1 + 120582119894) = (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
(33)
The other underlying inequality is 119890119909 ge 119890119909 and equalityholds if and only if 119909 = 1 we get
sum
120582119894gt0
119890120582
119894ge sum
120582119894gt0
119890120582119894= 119890 (120582
1+ 1205822sdot sdot sdot + 120582
119899119894) (34)
Substituting the inequalities (33) and (34) into (32)
119864119864 (119865119876119899) ge (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
+ 119890 (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (120582
1+ 1205822sdot sdot sdot + 120582
119899119894+ 120582119899119894+1
+ sdot sdot sdot + 120582119899)
+ (119890 minus 1) (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (119890 minus 1) (120582
1+ 1205822sdot sdot sdot + 120582
119899119894)
(35)
Note that
1205821+ 1205822sdot sdot sdot + 120582
119899119894=1
2119864 (119865119876
119899) (36)
From the above inequalities (35) and (36) we arrive at
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) le 119864119864 (119865119876
119899) (37)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleHence
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (38)
as desired
We now derive the upper bounds involving energy for theEstrada index of 119865119876
119899
Theorem 13 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 119864 (119865119876
119899) + 2119899minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(39)
Proof We consider that
119864119864 (119865119876119899) =
119899
sum
119894=1
119890120582
119894= 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896
le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
(40)
Taking into account the definition of graph energy equa-tion (1) we obtain
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) +
2119899
sum
119894=1
sum
119896ge2
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(41)
In light of the inequality (24) holds for integer 119896 ge 2 weobtain that
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(42)
Substituting (26) into (42) we get
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(43)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleFrom the above argument we get the result ofTheorem 13
6 Abstract and Applied Analysis
4 Conclusions
The main purpose of this paper is to investigate the Estradaindex of 119865119876
119899with 119899 ge 2 we established the explicit formulae
for calculating the Estrada index of 119865119876119899by deducing the
characteristic polynomial of the adjacency matrix in spectralgraph theory
Moreover some lower and upper bounds for Estradaindex of 119865119876
119899were proposed by utilizing the arithmetic and
geometric mean inequality The lower and upper bounds forthe Estrada index involving energy of119865119876
119899were also obtained
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework of Jia Bao Liu was supported by the Natural ScienceFoundation of Anhui Province of China under Grant noKJ2013B105 the work of Xiang-Feng Pan was supported bythe National Science Foundation of China under Grants10901001 11171097 and 11371028
References
[1] J M Xu Topological Strucure and Analysis of InterconnctionNetworks KluwerAcademicDordrechtTheNetherlands 2001
[2] I Gutman ldquoThe energy of a graphrdquo Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz vol 103 no22 pp 100ndash105 1978
[3] E Estrada ldquoCharacterization of 3Dmolecular structurerdquoChem-ical Physics Letters vol 319 no 5 pp 713ndash718 2000
[4] E Estrada and J A Rodrıguez-Velazquez ldquoSubgraph centralityin complex networksrdquo Physical Review E vol 71 no 5 ArticleID 056103 9 pages 2005
[5] E Estrada and J A Rodrıguez-Velazquez ldquoSpectral measures ofbipartivity in complex networksrdquo Physical Review E vol 72 no4 Article ID 046105 6 pages 2005
[6] K C Das I Gutman and B Zhou ldquoNew upper bounds onZagreb indicesrdquo Journal of Mathematical Chemistry vol 46 no2 pp 514ndash521 2009
[7] D M Cvetkovic M Doob I Gutman and A TorgasevRecent Results in the Theory of Graph Spectra vol 36 ofAnnals of Discrete Mathematics North-Holland AmsterdamThe Netherlands 1988
[8] E A Amawy and S Latifi ldquoProperties and performance offolded hypercubesrdquo IEEE Transactions on Parallel and Dis-tributed Systems vol 2 no 1 pp 31ndash42 1991
[9] R Indhumathi and S A Choudum ldquoEmbedding certainheight-balanced trees and complete 119901119898-ary trees into hyper-cubesrdquo Journal of Discrete Algorithms vol 22 pp 53ndash65 2013
[10] J Fink ldquoPerfect matchings extend to Hamilton cycles inhypercubesrdquo Journal of Combinatorial Theory B vol 97 no 6pp 1074ndash1076 2007
[11] J B Liu J D Cao X F Pan and A Elaiw ldquoThe Kirchhoffindex of hypercubes and related complex networksrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 5431897 pages 2013
[12] MChen andBXChen ldquoSpectra of foldedhypercubesrdquo Journalof East China Normal University (Nature Science) vol 2 pp 39ndash46 2011
[13] X He H Liu and Q Liu ldquoCycle embedding in faulty foldedhypercuberdquo International Journal of Applied Mathematics ampStatistics vol 37 no 7 pp 97ndash109 2013
[14] S Cao H Liu and X He ldquoOn constraint fault-free cycles infolded hypercuberdquo International Journal of AppliedMathematicsamp Statistics vol 42 no 12 pp 38ndash44 2013
[15] Y Zhang H Liu and M Liu ldquoCycles embedding on foldedhypercubes with vertex faultsrdquo International Journal of AppliedMathematics amp Statistics vol 41 no 11 pp 58ndash70 2013
[16] X B Chen ldquoConstruction of optimal independent spanningtrees on folded hypercubesrdquo Information Sciences vol 253 pp147ndash156 2013
[17] M Chen X Guo and S Zhai ldquoTotal chromatic number offolded hypercubesrdquo Ars Combinatoria vol 111 pp 265ndash2722013
[18] G H Wen Z S Duan W W Yu and G R Chen ldquoConsensusin multi-agent systems with communication constraintsrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 2pp 170ndash182 2012
[19] J B Liu X F Pan YWang and J D Cao ldquoThe Kirchhoff indexof folded hypercubes and some variant networksrdquoMathematicalProblems in Engineering vol 2014 Article ID 380874 9 pages2014
[20] E Estrada J A Rodrıguez-Velazquez and M Randic ldquoAtomicbranching in moleculesrdquo International Journal of QuantumChemistry vol 106 no 4 pp 823ndash832 2006
[21] B Zhou and Z Du ldquoSome lower bounds for Estrada indexrdquoIranian Journal of Mathematical Chemistry vol 1 no 2 pp 67ndash72 2010
[22] Y Shang ldquoLower bounds for the Estrada index of graphsrdquoElectronic Journal of Linear Algebra vol 23 pp 664ndash668 2012
[23] J P Liu and B L Liu ldquoBounds of the Estrada index of graphsrdquoApplied Mathematics vol 25 no 3 pp 325ndash330 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Notice that the spectral radius of 119865119876119899is 1205821= 119899 + 1 and
119903 = 119899 +1 applying Lemma6 we also give the lower and upperbounds connecting119864119864(119865119876
119899) and its spectral radius by simple
computations where the equality is impossible for any 119865119876119899
hence
119890119899+1
+ (2119899minus 1) 119890
(minus119899minus1)(2119899minus1)
lt 119864119864 (119865119876119899) lt 2119899minus 2 + 119890
119899+1+ 119890radic(119899+1)[2
119899minus(119899+1)]minus1
(30)
33 Some Properties on Estrada Index Involving Energy of119865119876119899 In this section we investigate the relations between the
Estrada index and the energy of 119865119876119899 We firstly prove the
lower bounds involving energy for the Estrada index of 119865119876119899
inTheorem 12 proof some techniques in [23] are referred to
Theorem 12 For any 119865119876119899with 119899 ge 2 one has
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (31)
Proof Assume that 119899119894denote the number of positive eigen-
values we begin with the definition of Estrada index119864119864(119865119876
119899)
119864119864 (119865119876119899) =
2119899
sum
119894=1
119890120582
119894= sum
120582119894le0
119890120582
119894+ sum
120582119894gt0
119890120582
119894 (32)
Since 119890119909 ge 1 + 119909 with equality holds if and only if 119909 = 0 wehave
sum
120582119894le0
119890120582
119894ge sum
120582119894le0
(1 + 120582119894) = (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
(33)
The other underlying inequality is 119890119909 ge 119890119909 and equalityholds if and only if 119909 = 1 we get
sum
120582119894gt0
119890120582
119894ge sum
120582119894gt0
119890120582119894= 119890 (120582
1+ 1205822sdot sdot sdot + 120582
119899119894) (34)
Substituting the inequalities (33) and (34) into (32)
119864119864 (119865119876119899) ge (2
119899minus 119899119894) + (120582
119899119894+1+ sdot sdot sdot + 120582
119899)
+ 119890 (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (120582
1+ 1205822sdot sdot sdot + 120582
119899119894+ 120582119899119894+1
+ sdot sdot sdot + 120582119899)
+ (119890 minus 1) (1205821+ 1205822sdot sdot sdot + 120582
119899119894)
= (2119899minus 119899119894) + (119890 minus 1) (120582
1+ 1205822sdot sdot sdot + 120582
119899119894)
(35)
Note that
1205821+ 1205822sdot sdot sdot + 120582
119899119894=1
2119864 (119865119876
119899) (36)
From the above inequalities (35) and (36) we arrive at
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) le 119864119864 (119865119876
119899) (37)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleHence
1
2(119890 minus 1) 119864 (119865119876
119899) + (2
119899minus 119899119894) lt 119864119864 (119865119876
119899) (38)
as desired
We now derive the upper bounds involving energy for theEstrada index of 119865119876
119899
Theorem 13 For any 119865119876119899with 119899 ge 2 one has
119864119864 (119865119876119899) lt 119864 (119865119876
119899) + 2119899minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(39)
Proof We consider that
119864119864 (119865119876119899) =
119899
sum
119894=1
119890120582
119894= 2119899+
2119899
sum
119894=1
sum
119896ge1
120582119896
119894
119896
le 2119899+
2119899
sum
119894=1
sum
119896ge1
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
(40)
Taking into account the definition of graph energy equa-tion (1) we obtain
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) +
2119899
sum
119894=1
sum
119896ge2
10038161003816100381610038161205821198941003816100381610038161003816
119896
119896
= 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896
2119899
sum
119894=1
[(120582119894)2
]1198962
(41)
In light of the inequality (24) holds for integer 119896 ge 2 weobtain that
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) + sum
119896ge2
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[
2119899
sum
119894=1
(120582119894)2
]
1198962
(42)
Substituting (26) into (42) we get
119864119864 (119865119876119899) le 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899
+ sum
119896ge0
1
119896[(119899 + 1) 2
119899]1198962
= 2119899+ 119864 (119865119876
119899) minus 1 minus radic(119899 + 1) 2119899 + 119890
radic(119899+1)2119899
(43)
with equality if and only if 119865119876119899is an empty graph with 2
119899
vertices which is impossibleFrom the above argument we get the result ofTheorem 13
6 Abstract and Applied Analysis
4 Conclusions
The main purpose of this paper is to investigate the Estradaindex of 119865119876
119899with 119899 ge 2 we established the explicit formulae
for calculating the Estrada index of 119865119876119899by deducing the
characteristic polynomial of the adjacency matrix in spectralgraph theory
Moreover some lower and upper bounds for Estradaindex of 119865119876
119899were proposed by utilizing the arithmetic and
geometric mean inequality The lower and upper bounds forthe Estrada index involving energy of119865119876
119899were also obtained
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework of Jia Bao Liu was supported by the Natural ScienceFoundation of Anhui Province of China under Grant noKJ2013B105 the work of Xiang-Feng Pan was supported bythe National Science Foundation of China under Grants10901001 11171097 and 11371028
References
[1] J M Xu Topological Strucure and Analysis of InterconnctionNetworks KluwerAcademicDordrechtTheNetherlands 2001
[2] I Gutman ldquoThe energy of a graphrdquo Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz vol 103 no22 pp 100ndash105 1978
[3] E Estrada ldquoCharacterization of 3Dmolecular structurerdquoChem-ical Physics Letters vol 319 no 5 pp 713ndash718 2000
[4] E Estrada and J A Rodrıguez-Velazquez ldquoSubgraph centralityin complex networksrdquo Physical Review E vol 71 no 5 ArticleID 056103 9 pages 2005
[5] E Estrada and J A Rodrıguez-Velazquez ldquoSpectral measures ofbipartivity in complex networksrdquo Physical Review E vol 72 no4 Article ID 046105 6 pages 2005
[6] K C Das I Gutman and B Zhou ldquoNew upper bounds onZagreb indicesrdquo Journal of Mathematical Chemistry vol 46 no2 pp 514ndash521 2009
[7] D M Cvetkovic M Doob I Gutman and A TorgasevRecent Results in the Theory of Graph Spectra vol 36 ofAnnals of Discrete Mathematics North-Holland AmsterdamThe Netherlands 1988
[8] E A Amawy and S Latifi ldquoProperties and performance offolded hypercubesrdquo IEEE Transactions on Parallel and Dis-tributed Systems vol 2 no 1 pp 31ndash42 1991
[9] R Indhumathi and S A Choudum ldquoEmbedding certainheight-balanced trees and complete 119901119898-ary trees into hyper-cubesrdquo Journal of Discrete Algorithms vol 22 pp 53ndash65 2013
[10] J Fink ldquoPerfect matchings extend to Hamilton cycles inhypercubesrdquo Journal of Combinatorial Theory B vol 97 no 6pp 1074ndash1076 2007
[11] J B Liu J D Cao X F Pan and A Elaiw ldquoThe Kirchhoffindex of hypercubes and related complex networksrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 5431897 pages 2013
[12] MChen andBXChen ldquoSpectra of foldedhypercubesrdquo Journalof East China Normal University (Nature Science) vol 2 pp 39ndash46 2011
[13] X He H Liu and Q Liu ldquoCycle embedding in faulty foldedhypercuberdquo International Journal of Applied Mathematics ampStatistics vol 37 no 7 pp 97ndash109 2013
[14] S Cao H Liu and X He ldquoOn constraint fault-free cycles infolded hypercuberdquo International Journal of AppliedMathematicsamp Statistics vol 42 no 12 pp 38ndash44 2013
[15] Y Zhang H Liu and M Liu ldquoCycles embedding on foldedhypercubes with vertex faultsrdquo International Journal of AppliedMathematics amp Statistics vol 41 no 11 pp 58ndash70 2013
[16] X B Chen ldquoConstruction of optimal independent spanningtrees on folded hypercubesrdquo Information Sciences vol 253 pp147ndash156 2013
[17] M Chen X Guo and S Zhai ldquoTotal chromatic number offolded hypercubesrdquo Ars Combinatoria vol 111 pp 265ndash2722013
[18] G H Wen Z S Duan W W Yu and G R Chen ldquoConsensusin multi-agent systems with communication constraintsrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 2pp 170ndash182 2012
[19] J B Liu X F Pan YWang and J D Cao ldquoThe Kirchhoff indexof folded hypercubes and some variant networksrdquoMathematicalProblems in Engineering vol 2014 Article ID 380874 9 pages2014
[20] E Estrada J A Rodrıguez-Velazquez and M Randic ldquoAtomicbranching in moleculesrdquo International Journal of QuantumChemistry vol 106 no 4 pp 823ndash832 2006
[21] B Zhou and Z Du ldquoSome lower bounds for Estrada indexrdquoIranian Journal of Mathematical Chemistry vol 1 no 2 pp 67ndash72 2010
[22] Y Shang ldquoLower bounds for the Estrada index of graphsrdquoElectronic Journal of Linear Algebra vol 23 pp 664ndash668 2012
[23] J P Liu and B L Liu ldquoBounds of the Estrada index of graphsrdquoApplied Mathematics vol 25 no 3 pp 325ndash330 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
4 Conclusions
The main purpose of this paper is to investigate the Estradaindex of 119865119876
119899with 119899 ge 2 we established the explicit formulae
for calculating the Estrada index of 119865119876119899by deducing the
characteristic polynomial of the adjacency matrix in spectralgraph theory
Moreover some lower and upper bounds for Estradaindex of 119865119876
119899were proposed by utilizing the arithmetic and
geometric mean inequality The lower and upper bounds forthe Estrada index involving energy of119865119876
119899were also obtained
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework of Jia Bao Liu was supported by the Natural ScienceFoundation of Anhui Province of China under Grant noKJ2013B105 the work of Xiang-Feng Pan was supported bythe National Science Foundation of China under Grants10901001 11171097 and 11371028
References
[1] J M Xu Topological Strucure and Analysis of InterconnctionNetworks KluwerAcademicDordrechtTheNetherlands 2001
[2] I Gutman ldquoThe energy of a graphrdquo Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz vol 103 no22 pp 100ndash105 1978
[3] E Estrada ldquoCharacterization of 3Dmolecular structurerdquoChem-ical Physics Letters vol 319 no 5 pp 713ndash718 2000
[4] E Estrada and J A Rodrıguez-Velazquez ldquoSubgraph centralityin complex networksrdquo Physical Review E vol 71 no 5 ArticleID 056103 9 pages 2005
[5] E Estrada and J A Rodrıguez-Velazquez ldquoSpectral measures ofbipartivity in complex networksrdquo Physical Review E vol 72 no4 Article ID 046105 6 pages 2005
[6] K C Das I Gutman and B Zhou ldquoNew upper bounds onZagreb indicesrdquo Journal of Mathematical Chemistry vol 46 no2 pp 514ndash521 2009
[7] D M Cvetkovic M Doob I Gutman and A TorgasevRecent Results in the Theory of Graph Spectra vol 36 ofAnnals of Discrete Mathematics North-Holland AmsterdamThe Netherlands 1988
[8] E A Amawy and S Latifi ldquoProperties and performance offolded hypercubesrdquo IEEE Transactions on Parallel and Dis-tributed Systems vol 2 no 1 pp 31ndash42 1991
[9] R Indhumathi and S A Choudum ldquoEmbedding certainheight-balanced trees and complete 119901119898-ary trees into hyper-cubesrdquo Journal of Discrete Algorithms vol 22 pp 53ndash65 2013
[10] J Fink ldquoPerfect matchings extend to Hamilton cycles inhypercubesrdquo Journal of Combinatorial Theory B vol 97 no 6pp 1074ndash1076 2007
[11] J B Liu J D Cao X F Pan and A Elaiw ldquoThe Kirchhoffindex of hypercubes and related complex networksrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 5431897 pages 2013
[12] MChen andBXChen ldquoSpectra of foldedhypercubesrdquo Journalof East China Normal University (Nature Science) vol 2 pp 39ndash46 2011
[13] X He H Liu and Q Liu ldquoCycle embedding in faulty foldedhypercuberdquo International Journal of Applied Mathematics ampStatistics vol 37 no 7 pp 97ndash109 2013
[14] S Cao H Liu and X He ldquoOn constraint fault-free cycles infolded hypercuberdquo International Journal of AppliedMathematicsamp Statistics vol 42 no 12 pp 38ndash44 2013
[15] Y Zhang H Liu and M Liu ldquoCycles embedding on foldedhypercubes with vertex faultsrdquo International Journal of AppliedMathematics amp Statistics vol 41 no 11 pp 58ndash70 2013
[16] X B Chen ldquoConstruction of optimal independent spanningtrees on folded hypercubesrdquo Information Sciences vol 253 pp147ndash156 2013
[17] M Chen X Guo and S Zhai ldquoTotal chromatic number offolded hypercubesrdquo Ars Combinatoria vol 111 pp 265ndash2722013
[18] G H Wen Z S Duan W W Yu and G R Chen ldquoConsensusin multi-agent systems with communication constraintsrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 2pp 170ndash182 2012
[19] J B Liu X F Pan YWang and J D Cao ldquoThe Kirchhoff indexof folded hypercubes and some variant networksrdquoMathematicalProblems in Engineering vol 2014 Article ID 380874 9 pages2014
[20] E Estrada J A Rodrıguez-Velazquez and M Randic ldquoAtomicbranching in moleculesrdquo International Journal of QuantumChemistry vol 106 no 4 pp 823ndash832 2006
[21] B Zhou and Z Du ldquoSome lower bounds for Estrada indexrdquoIranian Journal of Mathematical Chemistry vol 1 no 2 pp 67ndash72 2010
[22] Y Shang ldquoLower bounds for the Estrada index of graphsrdquoElectronic Journal of Linear Algebra vol 23 pp 664ndash668 2012
[23] J P Liu and B L Liu ldquoBounds of the Estrada index of graphsrdquoApplied Mathematics vol 25 no 3 pp 325ndash330 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Prediction of CBR Value from Index Properties of different ...web.uettaxila.edu.pk/techJournal/2017/No2/3-Prediction of CBR Value... · CBR value with Index properties of soils [v].](https://static.fdocuments.net/doc/165x107/5a710abb7f8b9a9d538c8b0f/prediction-of-cbr-value-from-index-properties-of-different-webuettaxilaedupktechjournal2017no23-prediction.jpg)
