Bounds of the extended Estrada index of graphs☆
Introduction
Throughout this paper we consider simple graphs, that are finite and undirected graphs without loops and multiple edges. Let be a graph with vertex set and edge set and let . Such a graph will be referred to as an (n, m)-graph. If the vertices vi and vj are adjacent, we write vivj ∈ E(G). For let di be the degree of the vertex vi.
The adjacency matrix of the graph G is defined so that its (i, j)-entry is equal to 1 if vivj ∈ E(G) and 0 otherwise. Let λ1 ≥ λ2 ≥ ⋅⋅⋅ ≥ λn denote the eigenvalues of A(G). The Estrada index [7] of G is defined as 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 was put forward by Yang et al. [18] and is defined so that its (i, j)-entry is equal to if vivj ∈ 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 Aex is a symmetric matrix of order n, all its eigenvalues are real. These are denoted by η1 ≥ η2 ≥ ⋅⋅⋅ ≥ ηn, and are known as the extended eigenvalues of the graph G. Since Aex 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 EEex(G) is defined as
where are the extended eigenvalues of the graph G.
As usual, by and 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.
Section snippets
Elementary properties of the extended Estrada index
Directly from the definition of the extended Estrada index, we conclude the following conclusions.
1. Denoting by the kth extended spectral moment of the graph G, and bearing in mind the power-series expansion of ex, we have As we know the kth spectral moment Mk(G) is equal to the number of self-returning walks of length k of the graph G [1]. For any edge vivj ∈ E(G), we define the edge weight ωij to be the (i, j)-entry of the matrix Aex(
Bounds for the extended Estrada index of (n,m)-graphs
Numerous lower and upper bounds for the Estrada index have been communicated. In what follows we show the lower and upper bounds for extended Estrada index.
Theorem 3.1 Let G be an (n, m)-graph. Then the extended Estrada index of G is bounded as
Equality on both sides of (3) holds if and only if.
Proof of the lower bound. Directly from the definition of the extended Estrada index, we get In view of the inequality between the arithmetic and
Bounds for the extended Estrada index of special graphs
Define an auxiliary quantity as
In view of the power-series expansion of ex and and bearing in mind the definition of the extended spectral moments, we have from which
It is known that is equal to the sum of the product of the edge weights in each self-returning walks of length k of the graph G. Consequently, for all graphs and all k ≥ 0. If the graph G is bipartite,
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2019, Applied Mathematics and ComputationCitation Excerpt :Since A(G) is a real symmetric matrix, all its eigenvalues are real. In paper [7], the authors gave many elementary properties of the extended Estrada index, which help us to understand the index better. In this paper, we use the algebraic method to show the extend Estrada index of some special graphs.
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