Isospectral graphs and the representation-theoretical spectrum

Abstract

A finite connected k-regular graph X, k greater than or equal to 3, determines the conjugacy class of a cocompact torsion-free lattice Gamma in the isometry group G of the universal covering tree. The associated quasi-regular representation L-2 (Gamma\G) of G can be considered as an a priori stronger notion of the spectrum of X, called the representation spectrum. We prove that two graphs as above are isospectral if and only if they are representation-isospectral. In other words, for a cocompact torsion-free lattice Gamma in G the spherical part of the spectrum of Gamma determines the whole spectrum. We give examples to show that this is not the case if the lattice has torsion. (C) 2004 Elsevier Ltd. All rights reserved.