Some properties of ergodicity coefficients with applications in spectral graph theory

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 {τ p }. 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 {τ p }. 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 τ p and Theorem 4.7 says that τ 1 is less than or equal to τ ∞ for the Laplacian matrix of every simple graph. Other discussions, open questions and examples are provided.