The second eigenvalue of regular graphs of given girth

Abstract

Lower bounds on the subdominant eigenvalue of regular graphs of given girth are derived. Our approach is to approximate the discrete spectrum of a finite regular graph by the continuous spectrum of an infinite regular tree. We interpret these spectra as probability distributions and the girth condition as equalities between the moments of these distributions. Then the associated orthogonal polynomials coincide up to a degree equal to half the girth, and their extremal zeroes provide bounds on the supports of these distributions.