Estrada Index and Laplacian Estrada Index of Random Interdependent Graphs

Yilun Shang

doi:10.3390/math8071063

Abstract

Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by E E ( G ) = ∑ i = 1 n e λ i ( A ( G ) ) and L E E ( G ) = ∑ i = 1 n e λ i ( L ( G ) ) , where { λ i ( A ( G ) ) } i = 1 n and { λ i ( L ( G ) ) } i = 1 n are the eigenvalues of its adjacency and Laplacian matrices, respectively. In this paper, we establish almost sure upper bounds and lower bounds for random interdependent graph model, which is fairly general encompassing Erdös-Rényi random graph, random multipartite graph, and even stochastic block model. Our results unravel the non-triviality of interdependent edges between different constituting subgraphs in spectral property of interdependent graphs.

Highlights

We consider a simple graph G = (V, E) on the vertex set V = {1, 2, · · ·, n} with |V | = n and the edge set E consisting of unordered pairs of vertices
Since G is undirected, both A( G ) and L( G ) are symmetric. It follows from algebraic graph theory (e.g., [1]) that A( G ) has n real eigenvalues arranged in the non-increasing order λ1 ( A( G )) ≥ λ2 ( A( G )) ≥ · · · ≥ λn ( A( G )), and L( G ) has n real and nonnegative eigenvalues ordered non-increasingly as λ1 ( L( G )) ≥ λ2 ( L( G )) ≥ · · · ≥ λn ( L( G )) = 0
In addition to fixed graphs, Estrada and Laplacian Estrada indices have been recently investigated for classical Erdös-Rényi random graph model as well as random multipartite graphs [14,15,16,17]

Summary

Introduction

In addition to fixed graphs, Estrada and Laplacian Estrada indices have been recently investigated for classical Erdös-Rényi random graph model as well as random multipartite graphs [14,15,16,17] These results are noteworthy in the sense that they contribute to the understanding of spectral theory of random networks and presenting estimates to EE and LEE for almost all graphs (as the number of vertices goes to infinity), which are typically much sharper than previous bounds for fixed graphs. We study the Estrada index and Laplacian Estrada index for the class of random interdependent graphs, which consist of m subgraphs with edges between different subgraphs appearing independently with probability p, where p ∈ (0, 1) is a constant.

Estrada Index of Random Interdependent Graphs

Laplacian Estrada Index of Random Interdependent Graphs

Conclusions