Bounds of the extended Estrada index of graphs

Abstract

Let G be a graph on n vertices and ${\eta }_1,{\eta }_2,\dots ,{\eta }_n$ the eigenvalues of its extended adjacency matrix. The extended Estrada index EE_ex is defined as the sum of the terms $e^{{\eta }_i},i=1,2,\dots ,n$. In this paper we establish lower and upper bounds for EE_ex in terms of the number of vertices and the number of edges and characterize the extremal graphs. Also the bounds for EE_ex of some special graphs are obtained.

Introduction

Throughout this paper we consider simple graphs, that are finite and undirected graphs without loops and multiple edges. Let $G=(V,E)$ be a graph with vertex set $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}$ and edge set $E=E\left(G\right),$ and let $\left|E\right(G\left)\right|=m$. Such a graph will be referred to as an (n, m)-graph. If the vertices v_i and v_j are adjacent, we write v_iv_j ∈ E(G). For $i=1,2,\dots ,n,$ let d_i be the degree of the vertex v_i.

The adjacency matrix $A=A\left(G\right)$ of the graph G is defined so that its (i, j)-entry is equal to 1 if v_iv_j ∈ E(G) and 0 otherwise. Let λ₁ ≥ λ₂ ≥ ⋅⋅⋅ ≥ λ_n denote the eigenvalues of A(G). The Estrada index [7] of G is defined as

$$EE\left(G\right)=\sum _{i=1}^n{e}^{{\lambda }_i}.$$

Since its introduction in 2000 as a molecular structure-descriptor, Estrada index has found various applications in many fields, such as biology, quantum chemistry and complex networks [7], [8], [9], [10], [11], [12]. In addition, this graph invariant has also received much attention from pure mathematicians for theoretical research. We refer the reader to the references [4], [5], [6], [13], [15], [16], [17], [19], [20]. For a recent survey on Estrada index, we refer to [14].

The extended adjacency matrix of the graph G , denoted by $A_{ex}=A_{ex}\left(G\right),$ was put forward by Yang et al. [18] and is defined so that its (i, j)-entry is equal to $\frac{1}{2}(\frac{d_i}{d_j}+\frac{d_j}{d_i})$ if v_iv_j ∈ E(G) and 0 otherwise. It is seen immediately that in the case of regular graphs, the extended adjacency matrix and the ordinary adjacency matrix coincide.

Since A_ex is a symmetric matrix of order n, all its eigenvalues are real. These are denoted by η₁ ≥ η₂ ≥ ⋅⋅⋅ ≥ η_n, and are known as the extended eigenvalues of the graph G. Since A_ex is a non-negative n × n matrix, its greatest eigenvalue may be viewed as the extended spectral radius of the graph G, a quantity first studied by Yang et al. [18].

Motivated by the Estrada index of graphs, here we introduce the extended Estrada index as follows:

Definition 1.1

For a graph G with n vertices, the extended Estrada index of G denoted by EE_ex(G) is defined as

$$E{E}_{ex}\left(G\right)=\sum _{i=1}^n{e}^{{\eta }_i},$$

where ${\eta }_1,{\eta }_2,\dots ,{\eta }_n$ are the extended eigenvalues of the graph G.

As usual, by $K_{p,q}(p+q=n),K_n,$ and ${\overline{K}}_n$ we denote, respectively, the complete bipartite graph, the complete graph, and the empty graph on n vertices. For other undefined notations and terminology from graph theory, the readers are referred to [2]. The rest of the paper is structured as follows. In Section 2, we state some elementary properties of the extended Estrada index. In Section 3 , we give some lower and upper bounds for the extended Estrada index of (n, m)-graphs and characterize the extremal graphs. In Section 4 , we give some lower and upper bounds for the extended Estrada index of some special graphs and characterize the extremal graphs.