# Some Properties on Estrada Index of Folded Hypercubes Networks

Authors: Jia-Bao Liu; Xiang-Feng Pan; Jinde Cao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

LetG 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.


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## Body

## 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 graphG=(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 λ1≥λ2≥⋯≥λn. In this work we are concerned with finite undirected connected simple graphs (networks). For the underlying graph theoretical definitions and notations we follow [1].The energy of the graphG [2] 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 [3], 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 byMk=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=0∞‍Mk(G)k!.At this point one should recall [4] 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]:
(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.For1≤i≤n, let di be the degree of vertex vi in G. The first Zagreb index [6] 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 [7].The hypercubesQn 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 [8–11]. As an important variant of Qn, the folded hypercubes networks FQn, proposed by Amawy and Latifi [8], 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 [12–19].The remainder of the present paper is organized as follows. In Section2, 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.LetPFQn(x) be the characteristic polynomial of the adjacency matrix of the folded hypercube FQn; the following results were shown in [12].Lemma 1 (see [12]).
The characteristic polynomial of the adjacency matrix of theFQn (n≥3) 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 [12]).
ForFQn with n≥3, the spectrum of adjacency matrix is as follows:
(1)Ifn≡0(mod 2),(7)Spec(FQn)=(-n+1-n+5-n+9⋯n-7n-3n+1Cn0+Cn1Cn2+Cn3Cn4+Cn5⋯Cnn-4+Cnn-3Cnn-2+Cnn-1Cnn),
(2) ifn≡1(mod 2),(8)Spec(FQn)=(-n-1-n+3-n+7⋯n-7n-3n+1Cn0Cn1+Cn2Cn3+Cn4⋯Cnn-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 [20]).
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 [21]).
LetG be a graph with m edges. For k≥4,
(10)Mk+2≥Mk,
with equality for all even k≥4 if and only if G consists of m copies of K2 and possibly isolated vertices and with equality for all odd k≥5 if and only if G is a bipartite graph.The following lemma is an immediate result of the previous lemma.Lemma 5 (see [22]).
LetG be an (n,m) graph with m edges. For k≥4,
(11)∑i=1n‍(2λi)k+2≥4∑i=1n‍(2λi)k,
with equality for all even k≥4 if and only if G consists of m copies of K2 and possibly isolated vertices and with equality for all odd k≥5 if and only if G is a bipartite graph.Lemma 6 (see [23]).
LetG be a regular graph of degree r≠0 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 [23]).
The Estrada indexEE(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 NetworksFQn

In this section, we present some explicit formulae for calculating the Estrada index ofFQn. For convenience, we assume that Cni=0 if i<0 or i>n.Theorem 8.
For anyFQn with n≥3, then(1)
EE(FQn)=∑i=0n/2‍(Cn2i+Cn2i+1)e4i-n+1,  i=0,1,…,n/2,  ifn≡0(mod 2);(2)
EE(FQn)=∑i=0n/2‍(Cn2i-1+Cn2i)e4i-n-1,  i=0,1,…,(n+1)/2,  ifn≡1(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 Lemma1, 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(1)
ifn≡0 (mod 2), FQn have  n/2+1 different eigenvalues 4i-n+1, with the multiplicities Cn2i+Cn2i+1, where i=0,1,…,n/2;(2)
ifn≡1 (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 Theorem8.


### 3.2. Some Bounds for the Estrada Index of Folded Hypercubes NetworksFQn

It is well known thatFQn have 2n vertices. Let λ1≥λ2≥⋯≥λn≥λn+1≥⋯≥λ2n 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 [22] are referred to.Theorem 9.
For anyFQn with n≥2, 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+2∑i<j‍eλieλj.
Noting thatM0=2n,  M1=0,  M2=(n+1)2n-1, and M3=6t, we obtain
(17)∑i=12n‍e2λi=∑i=12n‍∑k≥0‍(2λi)kk!=2n+(n+1)2n+1+8t+∑i=12n‍∑k≥4‍(2λi)kk!=2n+(n+1)2n+1+8t+∑k≥2‍1(2k)!∑i=12n‍(2λi)2k+∑k≥2‍1(2k+1)!∑i=12n‍(2λi)2k+1.
By Lemma5,
(18)∑i=1n‍(2λi)k+2≥4∑i=1n‍(2λi)k,
we can get that
(19)∑i=12n‍e2λi≥2n+(n+1)2n+1+8t+∑k≥2‍1(2k)!∑i=12n‍22k-4(2λi)4+∑k≥2‍1(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 terms2∑i<j‍eλieλj, by the arithmetic and geometric mean inequality and the fact that M1=0,
(20)2∑i<j‍eλieλj≥2n(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]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.
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 n≥2. Therefore, this implies the results of Theorem 9.We now consider the upper bound for the Estrada index ofFQn as follows.Theorem 10.
For anyFQn with n≥2, 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=12n‍∑k≥1‍λikk!≤2n+∑i=12n‍∑k≥1‍|λi|kk!=2n+∑k≥1‍1k!∑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+∑k≥1‍1k![∑i=12n‍(λi)2]k/2=2n-1+∑k≥0‍1k![∑i=12n‍(λi)2]k/2.
Since the equality holds inFQn,
(26)∑i=12n‍(λi)2=(n+1)2n.
Hence,
(27)EE(FQn)≤2n-1+∑k≥0‍1k![(n+1)2n]k/2=2n-1+∑k≥0‍(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 n≥2 that directly leads to the inequality in (27).
Hence, we can obtain the upper bound for the Estrada index ofFQn:
(28)EE(FQn)<2n-1+e(n+1)2n.
The proof of Theorem 10 is completed.Remark 11.
In [23], 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 ofFQn 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 ofFQn

In this section, we investigate the relations between the Estrada index and the energy ofFQn. We firstly prove the lower bounds involving energy for the Estrada index of FQn; in Theorem 12 proof, some techniques in [23] are referred to.Theorem 12.
For anyFQn with n≥2, one has
(31)12(e-1)E(FQn)+(2n-ni)<EE(FQn).Proof.
Assume thatni denote the number of positive eigenvalues; we begin with the definition of Estrada index EE(FQn):
(32)EE(FQn)=∑i=12n‍eiλ=∑λi≤0‍eiλ+∑λi>0‍eiλ.
Since ex≥1+x, with equality, holds if and only if x=0, we have
(33)∑λi≤0‍eiλ≥∑λi≤0‍(1+λi)=(2n-ni)+(λni+1+⋯+λn).
The other underlying inequality isex≥ex and equality holds if and only if x=1; we get
(34)∑λi>0‍eiλ≥∑λi>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 ofFQn.Theorem 13.
For anyFQn with n≥2, 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=12n‍∑k≥1‍λikk!≤2n+∑i=12n‍∑k≥1‍|λi|kk!.
Taking into account the definition of graph energy equation (1), we obtain
(41)EE(FQn)≤2n+E(FQn)+∑i=12n‍∑k≥2‍|λi|kk!=2n+E(FQn)+∑k≥2‍1k!∑i=12n‍[(λi)2]k/2.
In light of the inequality (24) holds for integer k≥2, we obtain that(42)EE(FQn)≤2n+E(FQn)+∑k≥2‍1k![∑i=12n‍(λi)2]k/2=2n+E(FQn)-1-(n+1)2n+∑k≥0‍1k![∑i=12n‍(λi)2]k/2.
Substituting (26) into (42), we get
(43)EE(FQn)≤2n+E(FQn)-1-(n+1)2n+∑k≥0‍1k![(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 Theorem13.


## 3.1. The Estrada Index of Folded Hypercubes NetworksFQn

In this section, we present some explicit formulae for calculating the Estrada index ofFQn. For convenience, we assume that Cni=0 if i<0 or i>n.Theorem 8.
For anyFQn with n≥3, then(1)
EE(FQn)=∑i=0n/2‍(Cn2i+Cn2i+1)e4i-n+1,  i=0,1,…,n/2,  ifn≡0(mod 2);(2)
EE(FQn)=∑i=0n/2‍(Cn2i-1+Cn2i)e4i-n-1,  i=0,1,…,(n+1)/2,  ifn≡1(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 Lemma1, 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(1)
ifn≡0 (mod 2), FQn have  n/2+1 different eigenvalues 4i-n+1, with the multiplicities Cn2i+Cn2i+1, where i=0,1,…,n/2;(2)
ifn≡1 (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 Theorem8.


## 3.2. Some Bounds for the Estrada Index of Folded Hypercubes NetworksFQn

It is well known thatFQn have 2n vertices. Let λ1≥λ2≥⋯≥λn≥λn+1≥⋯≥λ2n 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 [22] are referred to.Theorem 9.
For anyFQn with n≥2, 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+2∑i<j‍eλieλj.
Noting thatM0=2n,  M1=0,  M2=(n+1)2n-1, and M3=6t, we obtain
(17)∑i=12n‍e2λi=∑i=12n‍∑k≥0‍(2λi)kk!=2n+(n+1)2n+1+8t+∑i=12n‍∑k≥4‍(2λi)kk!=2n+(n+1)2n+1+8t+∑k≥2‍1(2k)!∑i=12n‍(2λi)2k+∑k≥2‍1(2k+1)!∑i=12n‍(2λi)2k+1.
By Lemma5,
(18)∑i=1n‍(2λi)k+2≥4∑i=1n‍(2λi)k,
we can get that
(19)∑i=12n‍e2λi≥2n+(n+1)2n+1+8t+∑k≥2‍1(2k)!∑i=12n‍22k-4(2λi)4+∑k≥2‍1(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 terms2∑i<j‍eλieλj, by the arithmetic and geometric mean inequality and the fact that M1=0,
(20)2∑i<j‍eλieλj≥2n(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]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.
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 n≥2. Therefore, this implies the results of Theorem 9.We now consider the upper bound for the Estrada index ofFQn as follows.Theorem 10.
For anyFQn with n≥2, 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=12n‍∑k≥1‍λikk!≤2n+∑i=12n‍∑k≥1‍|λi|kk!=2n+∑k≥1‍1k!∑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+∑k≥1‍1k![∑i=12n‍(λi)2]k/2=2n-1+∑k≥0‍1k![∑i=12n‍(λi)2]k/2.
Since the equality holds inFQn,
(26)∑i=12n‍(λi)2=(n+1)2n.
Hence,
(27)EE(FQn)≤2n-1+∑k≥0‍1k![(n+1)2n]k/2=2n-1+∑k≥0‍(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 n≥2 that directly leads to the inequality in (27).
Hence, we can obtain the upper bound for the Estrada index ofFQn:
(28)EE(FQn)<2n-1+e(n+1)2n.
The proof of Theorem 10 is completed.Remark 11.
In [23], 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 ofFQn 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 ofFQn

In this section, we investigate the relations between the Estrada index and the energy ofFQn. We firstly prove the lower bounds involving energy for the Estrada index of FQn; in Theorem 12 proof, some techniques in [23] are referred to.Theorem 12.
For anyFQn with n≥2, one has
(31)12(e-1)E(FQn)+(2n-ni)<EE(FQn).Proof.
Assume thatni denote the number of positive eigenvalues; we begin with the definition of Estrada index EE(FQn):
(32)EE(FQn)=∑i=12n‍eiλ=∑λi≤0‍eiλ+∑λi>0‍eiλ.
Since ex≥1+x, with equality, holds if and only if x=0, we have
(33)∑λi≤0‍eiλ≥∑λi≤0‍(1+λi)=(2n-ni)+(λni+1+⋯+λn).
The other underlying inequality isex≥ex and equality holds if and only if x=1; we get
(34)∑λi>0‍eiλ≥∑λi>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 ofFQn.Theorem 13.
For anyFQn with n≥2, 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=12n‍∑k≥1‍λikk!≤2n+∑i=12n‍∑k≥1‍|λi|kk!.
Taking into account the definition of graph energy equation (1), we obtain
(41)EE(FQn)≤2n+E(FQn)+∑i=12n‍∑k≥2‍|λi|kk!=2n+E(FQn)+∑k≥2‍1k!∑i=12n‍[(λi)2]k/2.
In light of the inequality (24) holds for integer k≥2, we obtain that(42)EE(FQn)≤2n+E(FQn)+∑k≥2‍1k![∑i=12n‍(λi)2]k/2=2n+E(FQn)-1-(n+1)2n+∑k≥0‍1k![∑i=12n‍(λi)2]k/2.
Substituting (26) into (42), we get
(43)EE(FQn)≤2n+E(FQn)-1-(n+1)2n+∑k≥0‍1k![(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 Theorem13.


## 4. Conclusions

The main purpose of this paper is to investigate the Estrada index ofFQn with n≥2; 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 ofFQn 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.



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*Source: 167623-2014-02-05.xml*