Spectra of Regular Graphs and Hypergraphs and Orthogonal Polynomials

In this paper we study the distribution of eigenvalues of regular graphs, regular hypergraphs, and biregular bipartite graphs of given girth by considering the polynomials orthogonal with respect to the measures attached to the spectra of such graphs and to the continuous spectra of their ‘universal covers’. Our estimates are tight for Biggs graphs and generalized polygons. We also give an application to the distribution of eigenvalues of Hecke operators acting on weight 2 cusp forms for certain congruence subgroups.