Lower Bounds for Gaussian Estrada Index of Graphs

Yilun Shang

doi:10.3390/sym10080325

Abstract

Suppose that G is a graph over n vertices. G has n eigenvalues (of adjacency matrix) represented by λ1,λ2,⋯,λn. The Gaussian Estrada index, denoted by H(G) (Estrada et al., Chaos 27(2017) 023109), can be defined as H(G)=∑i=1ne−λi2. Gaussian Estrada index underlines the eigenvalues close to zero, which plays an important role in chemistry reactions, such as molecular stability and molecular magnetic properties. In a network of particles governed by quantum mechanics, this graph-theoretic index is known to account for the information encoded in the eigenvalues of the Hamiltonian near zero by folding the graph spectrum. In this paper, we establish some new lower bounds for H(G) in terms of the number of vertices, the number of edges, as well as the first Zagreb index.

Highlights

Suppose that G is an undirected, simple graph containing n vertices and m edges
Estrada et al, [21] recently propose to extract key structural information hidden in the eigenvalues in proximity to zero in the spectra of networks by using a Gaussian matrix function. This novel method leads to the Gaussian Estrada index, H ( G ), characterized as follows: H = H ( G ) = Tr (e− A ) =
It is worth mentioning that in a network G of particles governed by the rules of quantum mechanics, the Gaussian Estrada index H can be viewed as the partition function of the system with Hamiltonian A2 based on the folded spectrum method [22]

Summary

Introduction

Suppose that G is an undirected, simple graph containing n vertices and m edges. Throughout the paper, we will refer to such a graph as an (n, m)-graph. The Estrada index of the graph G has been defined in [2,3,4,5,6,7] as n As a revealing graph-spectrum-based invariant, it has found numerous applications in chemistry, physics, and complex networks.

Results

Conclusion