A New Like Quantity Based on "Estrada Index"
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Research Article A New Like Quantity Based on “Estrada Index” ¨ or ¨ A. Dilek Gung Department of Mathematics, Science Faculty, Selc¸uk University, 42075 Konya, Turkey Correspondence should be addressed to A. Dilek Gung ¨ or, ¨ [email protected] Received 12 February 2010; Accepted 16 March 2010 Academic Editor: Martin Bohner Copyright q 2010 A. Dilek Gung ¨ or. ¨ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We first define a new Laplacian spectrum based on Estrada index, namely, Laplacian Estrada-like invariant, LEEL, and two new Estrada index-like quantities, denoted by S and EEX , respectively, that are generalized versions of the Estrada index. After that, we obtain some lower and upper bounds for LEEL, S, and EEX .
1. Introduction and Preliminaries It is known that, for an n, m-graph G i.e., an undirected graph with no loops and multiple edges, the numbers of vertices and edges of G are denoted by n and m, respectively. Throughout this paper, all graphs will be concerned as an n, m-graph. Let A AG be the adjacency matrix of G, and let λ1 , λ2 , . . . , λn be its eigenvalues. By 1, it is known that these eigenvalues form the spectrum of the graph G. Let G be connected graph on the vertex set V {v1 , v2 , . . . , vn }. Then the distance matrix D DG of G is defined as its i, j-entry is equal to dG vi , vj , denoted by dij , the distance in other words, the length of the shortest path between the vertices vi and vj of G. Let the eigenvalues of DG be ρ1 , ρ2 , . . . , ρn . Moreover let L LG be the Laplacian matrix of G formally it is denoted by LG DG − AG, and let μ1 , μ2 , . . . , μn be its eigenvalues. These eigenvalues form the Laplacian spectrum of the graph G see 2–4. Since AG, LG, and DG are real symmetric matrices, their eigenvalues are real numbers and so we can order them as λ1 ≥ λ2 ≥ · · · ≥ λn , μ1 ≥ μ2 ≥ · · · ≥ μn , and ρ1 ≥ ρ2 ≥ · · · ≥ ρn . These eigenvalues are shortly called A-eigenvalues, L-eigenvalues, and D-eigenvalues, respectively. The fundamental properties of graph eigenvalues can be found in the study in 1. Now we recall that the Estrada index of a simple connected graph G is defined by EE EEG
n e λi , i1
1.1
2
Journal of Inequalities and Applications
where, as depicted above, λ1 ≥ λ2 ≥ · · · ≥ λn are the A-eigenvalues see 5–8. Denoting by Mk Mk G the kth moment of the graph G, we get Mk Mk G ni1 λi k , and recalling the power-series expansion of ex , we have
EE
∞ Mk k0
k!
.
1.2
The Estrada index EE has an important role in Chemistry, since it is a proposed molecular structure descriptor, used in the modeling of certain features of the 3D structure of organic molecules, in particular of the degree of folding of proteins and other long-chain biopolymers. There exists a vast literature that studies Estrada index. For exampl
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