AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 167623 10.1155/2014/167623 167623 Research Article Some Properties on Estrada Index of Folded Hypercubes Networks Liu Jia-Bao 1, 2, 3 Pan Xiang-Feng 1 http://orcid.org/0000-0003-3133-7119 Cao Jinde 2, 4 Song Qiankun 1 School of Mathematics Science Anhui University Hefei 230601 China ahu.edu.cn 2 Department of Mathematics Southeast University Nanjing 210096 China seu.edu.cn 3 Anhui Xinhua University Hefei 230088 China axhu.edu.cn 4 Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia kau.edu.sa 2014 522014 2014 24 11 2013 19 12 2013 5 2 2014 2014 Copyright © 2014 Jia-Bao Liu et al. This 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 G be a simple graph with n vertices and let λ1,λ2,,λn be the eigenvalues of its adjacency matrix; the Estrada index EEG of the graph G is defined as the sum of the terms eλi,  i=1,2,,n. The n-dimensional folded hypercube networks FQn are an important and attractive variant of the n-dimensional hypercube networks Qn, which are obtained from Qn 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 FQn 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 FQn 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 G=(V,E), where V denotes the set of processors and E denotes the set of communication links between processors in networks. Let G be a graph with vertices labelled 1,2,,n. The adjacency matrix A(G) of G is an n×n matrix with the (i,j)-entry equal to 1 if vertices i and j are adjacent and 0 otherwise. The spectrum of G is the spectrum of its adjacency matrix and consists of the numbers λ1λ2λn. In this work we are concerned with finite undirected connected simple graphs (networks). For the underlying graph theoretical definitions and notations we follow .

The energy of the graph G  is defined as (1)E(G)=i=1n|λi|.

Another graph-spectrum-based invariant, recently put forward by Ernesto Estrada, is defined as (2)EE=EE(G)=i=1neλi. This graph invariant appeared for the first time in the year 2000, in a paper by Estrada , dealing with the folding of protein molecules. Estrada and Rodríguez-Velázquez showed that EE provides a measure of the centrality of complex (communication, social, metabolic, etc.) networks [4, 5].

Denote by Mk=Mk(G)=i=1n(λi)k the kth spectral moment of the graph G. From the Taylor expansion of ex, we have the following important relation between the Estrada index and the spectral moments of G: (3)EE(G)=k=0Mk(G)k!.

At this point one should recall  that Mk(G) is equal to the number of self-returning walks of length k of the graph G. The first few spectral moments of an (n,m)-graph with m edges and t triangles satisfy the following relations : (4)M0=i=1n(λi)0=n;M1=i=1n(λi)1=0;M2=i=1n(λi)2=2m;M3=i=1n(λi)3=6t.

For 1in, let di be the degree of vertex vi in G. The first Zagreb index  of the graph G is defined as Zg(G)=i=1ndi2: (5)M4=i=1n(λi)4=2Zg(G)-2m+8q;M5=i=1n(λi)5=30t+10p+10r, where p and q are the numbers of pentagons and quadrangles in G, and r is the number of subgraphs consisting of a triangle with a pendent vertex attached .

The hypercubes  Qn is one of the most popular and efficient interconnection networks due to its many excellent performances for some practical applications. There is a large amount of literature on the properties of hypercubes networks . As an important variant of Qn, the folded hypercubes networks FQn, proposed by Amawy and Latifi , are the graphs obtained from Qn by adding an edge between any pair of vertices complementary addresses. The folded hypercubes FQn obtained considerable attention due to its perfect properties, such as symmetry, regular structure, strong connectivity, small diameter, and many of its properties which have been explored .

The remainder of the present paper is organized as follows. In Section 2, we present some basic notations and some preliminaries in our discussion. The proofs of our main results are in Section 3 and some conclusions are given in Section 4, respectively.

2. Notations and Some Preliminaries

In this section, we introduce some basic properties which will be used in the proofs of our main results.

Let PFQn(x) be the characteristic polynomial of the adjacency matrix of the folded hypercube FQn; the following results were shown in .

Lemma 1 (see [<xref ref-type="bibr" rid="B9">12</xref>]).

The characteristic polynomial of the adjacency matrix of the FQn (n3) is (6)P(FQn;λ)=[λ-(n-7)][λ-(n-3)]3P(FQn-1;λ-1)×i=2n-2P(FQn-i;λ-(i-4)).

Lemma 2 (see [<xref ref-type="bibr" rid="B9">12</xref>]).

For FQn with n3, the spectrum of adjacency matrix is as follows:

(1)If n0 (mod 2),(7)Spec(FQn)=(-n+1-n+5-n+9n-7n-3n+1Cn0+Cn1Cn2+Cn3Cn4+Cn5Cnn-4+Cnn-3Cnn-2+Cnn-1Cnn),

(2) if n1 (mod 2),(8)Spec(FQn)=(-n-1-n+3-n+7n-7n-3n+1Cn0Cn1+Cn2Cn3+Cn4Cnn-4+Cnn-3Cnn-2+Cnn-1Cnn),where Cni are the binomial coefficients and the elements in the first and second rows are the eigenvalues of the adjacency matrix of FQn and the corresponding multiplicities, respectively.

Lemma 3 (see [<xref ref-type="bibr" rid="B6">20</xref>]).

The eigenvalues of a bipartite graph satisfy the pairing property: λn-i+1=λi,  i=1,2,,n. Therefore, if the graph G is bipartite and if η0 is nullity (the multiplicity of its eigenvalue zero), then (9)EE(G)=η0+2+cosh(i), where cosh stands for the hyperbolic cosine cosh(x)=(ex+e-x)/2, whereas +denotes summation over all positive eigenvalues of the corresponding graph.

Lemma 4 (see [<xref ref-type="bibr" rid="B10">21</xref>]).

Let G be a graph with m edges. For k4, (10)Mk+2Mk, with equality for all even k4 if and only if G consists of m copies of K2 and possibly isolated vertices and with equality for all odd k5 if and only if G is a bipartite graph.

The following lemma is an immediate result of the previous lemma.

Lemma 5 (see [<xref ref-type="bibr" rid="B11">22</xref>]).

Let G be an (n,m) graph with m edges. For k4, (11)i=1n(2λi)k+24i=1n(2λi)k, with equality for all even k4 if and only if G consists of m copies of K2 and possibly isolated vertices and with equality for all odd k5 if and only if G is a bipartite graph.

Lemma 6 (see [<xref ref-type="bibr" rid="B12">23</xref>]).

Let G be a regular graph of degree r0 and of order n. Then its Estrada index is bounded by (12)er+(n-1)e-r/(n-1)EE(G)<n-2+er+er(n-r)-1. Equality holds if and only if λ2=λ3==λn=-r/(n-1).

Lemma 7 (see [<xref ref-type="bibr" rid="B12">23</xref>]).

The Estrada index EE(G) and the graph energy E(G) satisfy the following inequality: (13)12E(G)(e-1)+n-n+EE(G)n-1+eE(G)/2, and equalities on both sides hold if and only if E(G)=0.

3. Main Results 3.1. The Estrada Index of Folded Hypercubes Networks <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M112"><mml:mi>F</mml:mi><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In this section, we present some explicit formulae for calculating the Estrada index of FQn. For convenience, we assume that Cni=0 if i<0 or i>n.

Theorem 8.

For any FQn with n3, then

EE(FQn)=i=0n/2(Cn2i+Cn2i+1)e4i-n+1,  i=0,1,,n/2,  ifn0  (mod 2);

EE(FQn)=i=0n/2(Cn2i-1+Cn2i)e4i-n-1,  i=0,1,,(n+1)/2,  ifn1 (mod 2),

where the 4i-n+1 and 4i-n-1(i=0,1,,n/2 or (n+1)/2) are the eigenvalues of the adjacent matrix of FQn and Cni denotes the binomial coefficients.

Proof.

By Lemma 1, the characteristic polynomial of the adjacent matrix of FQn is (14)P(FQn;λ)=[λ-(n-7)][λ-(n-3)]3P(FQn-1;λ-1)×i=2n-2P(FQn-i;λ-(i-4)).

Through calculating eigenvalues of characteristic polynomial and its multiplicities, we obtained that

if n0 (mod 2), FQn have  n/2+1 different eigenvalues 4i-n+1, with the multiplicities Cn2i+Cn2i+1, where i=0,1,,n/2;

if n1 (mod 2), FQn have (n+1)/2 different eigenvalues 4i-n-1, with the multiplicities Cn2i-1+Cn2i, where i=0,1,,(n+1)/2.

Combining with the definition of the Estrada index, we derived the result of Theorem 8.

3.2. Some Bounds for the Estrada Index of Folded Hypercubes Networks <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M144"><mml:mi>F</mml:mi><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

It is well known that FQn have 2n vertices. Let λ1λ2λnλn+1λ2n be the eigenvalues of FQn with nonincreasing order. In order to obtain the bounds for the Estrada index of FQn, we prove some results by utilizing the arithmetic and geometric mean inequality; in our proof, some techniques in  are referred to.

Theorem 9.

For any FQn with n2, one has (15)4n+(n+1)2n+1+8t+[cosh(2)-3]M4+[cosh(2)-103]M5<EE(FQn), where M4=2Zg(G)-2m+8q,  M5=30t+10p+10r,  p and q are the numbers of pentagons and quadrangles in FQn, and r is the number of subgraphs consisting of a triangle with a pendent vertex attached.

Proof.

In order to obtain the lower bounds for the Estrada index, consider that (16)EE2(FQn)=i=12ne2λi+2i<jeλieλj.

Noting that M0=2n,  M1=0,  M2=(n+1)2n-1, and M3=6t, we obtain (17)i=12ne2λi=i=12nk0(2λi)kk!=2n+(n+1)2n+1+8t+i=12nk4(2λi)kk!=2n+(n+1)2n+1+8t+k21(2k)!i=12n(2λi)2k+k21(2k+1)!i=12n(2λi)2k+1.

By Lemma 5, (18)i=1n(2λi)k+24i=1n(2λi)k, we can get that (19)i=12ne2λi2n+(n+1)2n+1+8t+k21(2k)!i=12n22k-4(2λi)4+k21(2k+1)!i=12n22k-4(2λi)5=2n+(n+1)2n+1+8t+[cosh(2)-3]M4+[cosh(2)-103]M5, where M4=2Zg(G)-2m+8q,  M5=30t+10p+10r,  p and q are the numbers of pentagons and quadrangles in FQn, and r is the number of subgraphs consisting of a triangle with a pendent vertex attached.

As for the terms 2i<jeλieλj, by the arithmetic and geometric mean inequality and the fact that M1=0, (20)2i<jeλieλj2n(2n-1)(i<jeλieλj)2/2n(2n-1)=2n(2n-1)[(i=1eλi)2n-1]2/2n(2n-1)=2n(2n-1)(eM1)2/2n=2n(2n-1), where the equality holds if and only if λ1==λ2n.

Combining with equalities (19) and (20), (21)4n+(n+1)2n+1+8t+[cosh(2)-3]M4+[cosh(2)-103]M5EE(FQn), where M4=2Zg(G)-2m+8q,  M5=30t+10p+10r,p and q are the numbers of pentagons and quadrangles in FQn, and r is the number of subgraphs consisting of a triangle with a pendent vertex attached.

Notice that the equality of (21) holds if and only if the equalities of (19) and (20) hold; that is, the equality holds if and only if λ1==λ2n, which is impossible for any FQn with n2. Therefore, this implies the results of Theorem 9.

We now consider the upper bound for the Estrada index of FQn as follows.

Theorem 10.

For any FQn with n2, one has (22)EE(FQn)<2n-1+e(n+1)2n.

Proof.

According to the definition of Estrada index we get (23)EE(FQn)=2n+i=12nk1λikk!2n+i=12nk1|λi|kk!=2n+k11k!i=12n[(λi)2]k/2.

Notice the inequality (24)i=12n[(λi)2]k/2[i=12n(λi)2]k/2; substituting inequality (24) into (23) we obtain that (25)EE(FQn)2n+k11k![i=12n(λi)2]k/2=2n-1+k01k![i=12n(λi)2]k/2.

Since the equality holds in FQn, (26)i=12n(λi)2=(n+1)2n. Hence, (27)EE(FQn)2n-1+k01k![(n+1)2n]k/2=2n-1+k0(n+1)2nkk!=2n-1+e(n+1)2n. It is evident that equality of (25) will be attained if and only if the graph FQn has no nonzero eigenvalues, which, in turn, happens only in the case of the edgeless graph Kn¯; it is impossible for any FQn with n2 that directly leads to the inequality in (27).

Hence, we can obtain the upper bound for the Estrada index of FQn: (28)EE(FQn)<2n-1+e(n+1)2n. The proof of Theorem 10 is completed.

Remark 11.

In , it was proved that (29)er+(n-1)e-r/(n-1)EE(G)<n-2+er+er(n-r)-1, with equality, holds if and only if λ2=λ3==λn=-r/(n-1).

Notice that the spectral radius of FQn is λ1=n+1 and r=n+1; applying Lemma 6, we also give the lower and upper bounds connecting EE(FQn) and its spectral radius by simple computations, where the equality is impossible for any FQn; hence (30)en+1+(2n-1)e(-n-1)/(2n-1)<EE(FQn)<2n-2+en+1+e(n+1)[2n-(n+1)]-1.

3.3. Some Properties on Estrada Index Involving Energy of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M211"><mml:mi>F</mml:mi><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In this section, we investigate the relations between the Estrada index and the energy of FQn. We firstly prove the lower bounds involving energy for the Estrada index of FQn; in Theorem 12 proof, some techniques in  are referred to.

Theorem 12.

For any FQn with n2, one has (31)12(e-1)E(FQn)+(2n-ni)<EE(FQn).

Proof.

Assume that ni denote the number of positive eigenvalues; we begin with the definition of Estrada index EE(FQn): (32)EE(FQn)=i=12neiλ=λi0eiλ+λi>0eiλ. Since ex1+x, with equality, holds if and only if x=0, we have (33)λi0eiλλi0(1+λi)=(2n-ni)+(λni+1++λn).

The other underlying inequality is exex and equality holds if and only if x=1; we get (34)λi>0eiλλi>0eλi=e(λ1+λ2+λni).

Substituting the inequalities (33) and (34) into (32), (35)EE(FQn)(2n-ni)+(λni+1++λn)+e(λ1+λ2+λni)=(2n-ni)+(λ1+λ2+λni+λni+1++λn)+(e-1)(λ1+λ2+λni)=(2n-ni)+(e-1)(λ1+λ2+λni).

Note that (36)λ1+λ2+λni=12E(FQn).

From the above inequalities (35) and (36), we arrive at (37)12(e-1)E(FQn)+(2n-ni)EE(FQn), with equality if and only if FQn is an empty graph with 2n vertices, which is impossible.

Hence, (38)12(e-1)E(FQn)+(2n-ni)<EE(FQn), as desired.

We now derive the upper bounds involving energy for the Estrada index of FQn.

Theorem 13.

For any FQn with n2, one has (39)EE(FQn)<E(FQn)+2n-1-(n+1)2n+e(n+1)2n.

Proof.

We consider that (40)EE(FQn)=i=1neiλ=2n+i=12nk1λikk!2n+i=12nk1|λi|kk!.

Taking into account the definition of graph energy equation (1), we obtain (41)EE(FQn)2n+E(FQn)+i=12nk2|λi|kk!=2n+E(FQn)+k21k!i=12n[(λi)2]k/2.

In light of the inequality (24) holds for integer k2, we obtain that(42)EE(FQn)2n+E(FQn)+k21k![i=12n(λi)2]k/2=2n+E(FQn)-1-(n+1)2n+k01k![i=12n(λi)2]k/2.

Substituting (26) into (42), we get (43)EE(FQn)2n+E(FQn)-1-(n+1)2n+k01k![(n+1)2n]k/2=2n+E(FQn)-1-(n+1)2n+e(n+1)2n, with equality if and only if FQn is an empty graph with 2n vertices, which is impossible.

From the above argument, we get the result of Theorem 13.

4. Conclusions

The main purpose of this paper is to investigate the Estrada index of FQn with n2; we established the explicit formulae for calculating the Estrada index of FQn by deducing the characteristic polynomial of the adjacency matrix in spectral graph theory.

Moreover, some lower and upper bounds for Estrada index of FQn were proposed by utilizing the arithmetic and geometric mean inequality. The lower and upper bounds for the Estrada index involving energy of FQn were also obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work of Jia Bao Liu was supported by the Natural Science Foundation of Anhui Province of China under Grant no. KJ2013B105; the work of Xiang-Feng Pan was supported by the National Science Foundation of China under Grants 10901001, 11171097, and 11371028.

Xu J. M. Topological Strucure and Analysis of Interconnction Networks 2001 Dordrecht, The Netherlands Kluwer Academic Gutman I. The energy of a graph Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz 1978 103 22 100 105 MR525890 Estrada E. Characterization of 3D molecular structure Chemical Physics Letters 2000 319 5 713 718 10.1016/S0009-2614(00)00158-5 Estrada E. Rodríguez-Velázquez J. A. Subgraph centrality in complex networks Physical Review E 2005 71 5 9 056103 10.1103/PhysRevE.71.056103 MR2171832 Estrada E. Rodríguez-Velázquez J. A. Spectral measures of bipartivity in complex networks Physical Review E 2005 72 4 6 046105 10.1103/PhysRevE.72.046105 MR2202758 Das K. C. Gutman I. Zhou B. New upper bounds on Zagreb indices Journal of Mathematical Chemistry 2009 46 2 514 521 10.1007/s10910-008-9475-3 MR2583047 Cvetković D. M. Doob M. Gutman I. Torgašev A. Recent Results in the Theory of Graph Spectra 1988 36 Amsterdam, The Netherlands North-Holland Annals of Discrete Mathematics MR926481 Amawy E. A. Latifi S. Properties and performance of folded hypercubes IEEE Transactions on Parallel and Distributed Systems 1991 2 1 31 42 10.1109/71.80187 Indhumathi R. Choudum S. A. Embedding certain height-balanced trees and complete pm-ary trees into hypercubes Journal of Discrete Algorithms 2013 22 53 65 10.1016/j.jda.2013.07.005 MR3096449 Fink J. Perfect matchings extend to Hamilton cycles in hypercubes Journal of Combinatorial Theory B 2007 97 6 1074 1076 10.1016/j.jctb.2007.02.007 MR2354719 Liu J. B. Cao J. D. Pan X. F. Elaiw A. The Kirchhoff index of hypercubes and related complex networks Discrete Dynamics in Nature and Society 2013 2013 7 543189 10.1155/2013/543189 Chen M. Chen B. X. Spectra of folded hypercubes Journal of East China Normal University (Nature Science) 2011 2 39 46 MR2866653 He X. Liu H. Liu Q. Cycle embedding in faulty folded hypercube International Journal of Applied Mathematics & Statistics 2013 37 7 97 109 MR3066475 Cao S. Liu H. He X. On constraint fault-free cycles in folded hypercube International Journal of Applied Mathematics & Statistics 2013 42 12 38 44 MR3093304 Zhang Y. Liu H. Liu M. Cycles embedding on folded hypercubes with vertex faults International Journal of Applied Mathematics & Statistics 2013 41 11 58 70 MR3065344 Chen X. B. Construction of optimal independent spanning trees on folded hypercubes Information Sciences 2013 253 147 156 10.1016/j.ins.2013.07.016 MR3107398 Chen M. Guo X. Zhai S. Total chromatic number of folded hypercubes Ars Combinatoria 2013 111 265 272 MR3100178 Wen G. H. Duan Z. S. Yu W. W. Chen G. R. Consensus in multi-agent systems with communication constraints International Journal of Robust and Nonlinear Control 2012 22 2 170 182 10.1002/rnc.1687 MR2875190 Liu J. B. Pan X. F. Wang Y. Cao J. D. The Kirchhoff index of folded hypercubes and some variant networks Mathematical Problems in Engineering 2014 2014 9 380874 10.1155/2014/380874 Estrada E. Rodríguez-Velázquez J. A. Randić M. Atomic branching in molecules International Journal of Quantum Chemistry 2006 106 4 823 832 10.1002/qua.20850 Zhou B. Du Z. Some lower bounds for Estrada index Iranian Journal of Mathematical Chemistry 2010 1 2 67 72 Shang Y. Lower bounds for the Estrada index of graphs Electronic Journal of Linear Algebra 2012 23 664 668 MR2966798 Liu J. P. Liu B. L. Bounds of the Estrada index of graphs Applied Mathematics 2010 25 3 325 330 10.1007/s11766-010-2237-6 MR2679351