Some Properties on Estrada Index of Folded Hypercubes Networks

Abstract

Let be a simple graph with vertices and let be the eigenvalues of its adjacency matrix; the Estrada index of the graph is defined as the sum of the terms ,  . The -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 . The adjacency matrix of is an matrix with the -entry equal to 1 if vertices and are adjacent and 0 otherwise. The spectrum of is the spectrum of its adjacency matrix and consists of the numbers . 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 graph [2] is defined as

Another graph-spectrum-based invariant, recently put forward by Ernesto Estrada, is defined as 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 provides a measure of the centrality of complex (communication, social, metabolic, etc.) networks [4, 5].

Denote by 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 :

At this point one should recall [4] that is equal to the number of self-returning walks of length of the graph . The first few spectral moments of an -graph with edges and triangles satisfy the following relations [4]:

For , let be the degree of vertex in . The first Zagreb index [6] of the graph is defined as : where and are the numbers of pentagons and quadrangles in , and is the number of subgraphs consisting of a triangle with a pendent vertex attached [7].

The hypercubes   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 , the folded hypercubes networks , proposed by Amawy and Latifi [8], are the graphs obtained from by adding an edge between any pair of vertices complementary addresses. The folded hypercubes 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 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 be the characteristic polynomial of the adjacency matrix of the folded hypercube ; the following results were shown in [12].

Lemma 1 (see [12]). The characteristic polynomial of the adjacency matrix of the () is

Lemma 2 (see [12]). For with , the spectrum of adjacency matrix is as follows:
(1)If (mod 2),
(2) if (mod 2), where are the binomial coefficients and the elements in the first and second rows are the eigenvalues of the adjacency matrix of and the corresponding multiplicities, respectively.

Lemma 3 (see [20]). The eigenvalues of a bipartite graph satisfy the pairing property: ,  . Therefore, if the graph is bipartite and if is nullity (the multiplicity of its eigenvalue zero), then where stands for the hyperbolic cosine , whereas denotes summation over all positive eigenvalues of the corresponding graph.

Lemma 4 (see [21]). Let be a graph with edges. For , with equality for all even if and only if consists of copies of and possibly isolated vertices and with equality for all odd if and only if is a bipartite graph.

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

Lemma 5 (see [22]). Let be an graph with edges. For , with equality for all even if and only if consists of copies of and possibly isolated vertices and with equality for all odd if and only if is a bipartite graph.

Lemma 6 (see [23]). Let be a regular graph of degree and of order . Then its Estrada index is bounded by Equality holds if and only if .

Lemma 7 (see [23]). The Estrada index and the graph energy satisfy the following inequality: and equalities on both sides hold if and only if .

3. Main Results

3.1. The Estrada Index of Folded Hypercubes Networks

In this section, we present some explicit formulae for calculating the Estrada index of . For convenience, we assume that if or .

Theorem 8. For any with , then(1),  ,  if  (mod 2);(2),  ,  if (mod 2), where the and or are the eigenvalues of the adjacent matrix of and denotes the binomial coefficients.

Proof. By Lemma 1, the characteristic polynomial of the adjacent matrix of is
Through calculating eigenvalues of characteristic polynomial and its multiplicities, we obtained that(1)if (mod 2), have   different eigenvalues , with the multiplicities , where ;(2)if (mod 2), have different eigenvalues , with the multiplicities , where .
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

It is well known that have vertices. Let be the eigenvalues of with nonincreasing order. In order to obtain the bounds for the Estrada index of , 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 any with , one has where ,  ,   and are the numbers of pentagons and quadrangles in , and 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
Noting that ,  ,  , and , we obtain
By Lemma 5, we can get that where ,  ,   and are the numbers of pentagons and quadrangles in , and is the number of subgraphs consisting of a triangle with a pendent vertex attached.
As for the terms , by the arithmetic and geometric mean inequality and the fact that , where the equality holds if and only if .
Combining with equalities (19) and (20), where ,  , and are the numbers of pentagons and quadrangles in , and 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 , which is impossible for any with . Therefore, this implies the results of Theorem 9.

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

Theorem 10. For any with , one has

Proof. According to the definition of Estrada index we get
Notice the inequality substituting inequality (24) into (23) we obtain that
Since the equality holds in , Hence, It is evident that equality of (25) will be attained if and only if the graph has no nonzero eigenvalues, which, in turn, happens only in the case of the edgeless graph ; it is impossible for any with that directly leads to the inequality in (27).
Hence, we can obtain the upper bound for the Estrada index of : The proof of Theorem 10 is completed.

Remark 11. In [23], it was proved that with equality, holds if and only if .

Notice that the spectral radius of is and ; applying Lemma 6, we also give the lower and upper bounds connecting and its spectral radius by simple computations, where the equality is impossible for any ; hence

3.3. Some Properties on Estrada Index Involving Energy of

In this section, we investigate the relations between the Estrada index and the energy of . We firstly prove the lower bounds involving energy for the Estrada index of ; in Theorem 12 proof, some techniques in [23] are referred to.

Theorem 12. For any with , one has

Proof. Assume that denote the number of positive eigenvalues; we begin with the definition of Estrada index : Since , with equality, holds if and only if , we have
The other underlying inequality is and equality holds if and only if ; we get
Substituting the inequalities (33) and (34) into (32),
Note that
From the above inequalities (35) and (36), we arrive at with equality if and only if is an empty graph with vertices, which is impossible.
Hence, as desired.

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

Theorem 13. For any with , one has

Proof. We consider that
Taking into account the definition of graph energy equation (1), we obtain
In light of the inequality (24) holds for integer , we obtain that
Substituting (26) into (42), we get with equality if and only if is an empty graph with 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 with ; we established the explicit formulae for calculating the Estrada index of by deducing the characteristic polynomial of the adjacency matrix in spectral graph theory.

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