Some properties of ergodicity coefficients with applications in spectral graph theory

TL;DR

This paper introduces bounds on eigenvalues of constant row-sum matrices using ergodicity coefficients and applies these results to spectral graph theory, including Laplacian matrices and graph connectivity.

Contribution

It provides new bounds and approximation methods for eigenvalues of matrices using ergodicity coefficients, with applications to spectral graph theory.

Findings

Bounds on the largest non-trivial eigenvalue of matrices.

Bounds on the smallest non-trivial eigenvalue for nonsingular matrices.

Applications to spectral radius and algebraic connectivity of graphs.

Abstract

The main result is Corollary 2.9 which provides upper bounds on, and even better, approximates the largest non-trivial eigenvalue in absolute value of real constant row-sum matrices by the use of vector norm based ergodicity coefficients Tp. If the constant row-sum matrix is nonsingular, then it is also shown how its smallest non-trivial eigenvalue in absolute value can be bounded by using Tp. In the last section, these two results are applied to bound the spectral radius of the Laplacian matrix as well as the algebraic connectivity of its associated graph. Many other results are obtained. In particular, Theorem 2.15 is a convergence theorem for Tp and Theorem 4.7 compares some ergodicity coefficients to each other.