Lower Bounds for Gaussian Estrada Index of Graphs

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.