^{1, 2, 3}

^{1}

^{2, 4}

^{1}

^{2}

^{3}

^{4}

Let

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

The energy of the graph

Another graph-spectrum-based invariant, recently put forward by Ernesto Estrada, is defined as

Denote by

At this point one should recall [

For

The hypercubes

The remainder of the present paper is organized as follows. In Section

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

Let

The characteristic polynomial of the adjacency matrix of the

For

(1)If

(2) if

The eigenvalues of a bipartite graph satisfy the pairing property:

Let

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

Let

Let

The Estrada index

In this section, we present some explicit formulae for calculating the Estrada index of

For any

By Lemma

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

if

if

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

It is well known that

For any

In order to obtain the lower bounds for the Estrada index, consider that

Noting that

By Lemma

As for the terms

Combining with equalities (

Notice that the equality of (

We now consider the upper bound for the Estrada index of

For any

According to the definition of Estrada index we get

Notice the inequality

Since the equality holds in

Hence, we can obtain the upper bound for the Estrada index of

In [

Notice that the spectral radius of

In this section, we investigate the relations between the Estrada index and the energy of

For any

Assume that

The other underlying inequality is

Substituting the inequalities (

Note that

From the above inequalities (

Hence,

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

For any

We consider that

Taking into account the definition of graph energy equation (

In light of the inequality (

Substituting (

From the above argument, we get the result of Theorem

The main purpose of this paper is to investigate the Estrada index of

Moreover, some lower and upper bounds for Estrada index of

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

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.