Lower bounds for the Estrada index using mixing time and Laplacian spectrum

  title={Lower bounds for the Estrada index using mixing time and Laplacian spectrum},
  author={Yilun Shang},
  journal={Rocky Mountain Journal of Mathematics},
  • Y. Shang
  • Published 1 December 2013
  • Mathematics
  • Rocky Mountain Journal of Mathematics
The logarithm of the Estrada index has been recently proposed as a spectral measure to characterize the robustness of complex networks. We derive novel analytic lower bounds for the logarithm of the Estrada index based on the Laplacian spectrum and the mixing times of random walks on the network. The main techniques employed are some inequalities, such as the thermodynamic inequality in statistical mechanics, a trace inequality of von Neumann, and a refined harmonic-arithmetic mean inequality. 
Laplacian Estrada and Normalized Laplacian Estrada Indices of Evolving Graphs
It is found that neither the static snapshot graphs nor the aggregated graph can approximate the evolving graph itself, indicating the fundamental difference between the static and dynamic Estrada indices.
The many facets of the Estrada indices of graphs and networks
  • E. Estrada
  • Mathematics, Computer Science
    SeMA Journal
  • 2021
This article formalizes many of the Estrada indices proposed and studied in the mathematical literature serving as a guide for their further studies and analyses many of them computationally for small graphs as well as large complex networks.
More on the normalized Laplacian Estrada index
Let $G$ be a simple graph of order $N$. The normalized Laplacian Estrada index of $G$ is defined as $NEE(G)=\sum_{i=1}^Ne^{\lambda_i}$, where $\lambda_1,\lambda_2,\cdots,\lambda_N$ are the normalized
The Estrada index of evolving graphs
  • Y. Shang
  • Mathematics
    Appl. Math. Comput.
  • 2015
Estimating the distance Estrada index
Suppose $G$ is a simple graph on $n$ vertices. The $D$-eigenvalues$\mu_1,\mu_2,\cdots,\mu_n$ of $G$ are the eigenvalues of itsdistance matrix. The distance Estrada index of $G$ is defined
Estrada Index of Random Bipartite Graphs
Upper and lower bounds of \(EE(G)\) for almost all bipartite graphs are given by investigating the upper and lower limits of the spectrum of random matrices.
Estrada and L-Estrada Indices of Edge-Independent Random Graphs
Upper and lower bounds to \(EE\) and \(\mathcal{L}EE\) are established for edge-independent random graphs, containing the classical Erdos-Renyi graphs as special cases.
On the spectrum of linear dependence graph of a finite dimensional vector space
The main contribution of this article is to find eigen values of adjacency matrix, Laplacian matrix and distance matrix of this graph, which is Eulerian if and only if q is odd.
On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n
Given a commutative ring R with identity 1 ≠ 0 , let the set Z ( R ) denote the set of zero-divisors and let Z * ( R ) = Z ( R ) ∖ { 0 } be the set of non-zero zero-divisors of R. The zero-divisor


The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla- cian eigenvalue 2 and its relation to
Estrada Index of Random Graphs
The Estrada index of a graph G of order n is defined as EE(G )= � n=1 e λ i , where λ1 ,λ 2 ,...,λ n are the eigenvalues of the graph G. By the limiting behavior of the spectrum of random symmetric
The Kemeny Constant for Finite Homogeneous Ergodic Markov Chains
A new formula for the Kemeny constant is presented and several perturbation results for the constant are developed, including conditions under which it is a convex function and for chains whose transition matrix has a certain directed graph structure.
Random Walks on Graphs: A Survey
Estimates on the important parameters of access time, commute time, cover time and mixing time are discussed and recent algorithmic applications of random walks are sketched, in particular to the problem of sampling.
Spectral measures of bipartivity in complex networks.
It is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks, trophic interactions in food webs, construction principles in metabolic networks, or communities in social networks.
Biased edge failure in scale-free networks based on natural connectivity
The natural connectivity is recently reported as a novel spectral measure of robustness in complex networks. It has a clear physical meaning and a simple mathematical formulation. In this article,
Local Natural Connectivity in Complex Networks
It is shown that an inequality for eigenvalues of Erdos-Renyi random graphs can be derived from an induced subgraph of it based on interlacing theorems and this work decomposes the natural connectivity of a network as localnatural connectivity of its connected components.
Spectral Graph Theory
Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues
The Structure and Function of Complex Networks
Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.