Measure-theoretic bounds on the spectral radius of graphs from walks

Abstract

Let G be an undirected graph with adjacency matrix A and spectral radius ρ. Let wk,ϕk and ϕk(i) be, respectively, the number walks of length k, closed walks of length k and closed walks starting and ending at vertex i after k steps. In this paper, we propose a measure-theoretic framework which allows us to relate walks in a graph with its spectral properties. In particular, we show that wk,ϕk and ϕk(i) can be interpreted as the moments of three different measures, all of them supported on the spectrum of A. Building on this interpretation, we leverage results from the classical moment problem to formulate a hierarchy of new lower and upper bounds on ρ, as well as provide alternative proofs to several well-known bounds in the literature.