Isospectral graphs and the representation-theoretical spectrum

A finite connected k-regular graph X,k≥3, determines the conjugacy class of a cocompact torsion-free lattice Γ in the isometry group G of the universal covering tree. The associated quasi-regular representation L2(Γ ⧹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 Γ in G the spherical part of the spectrum of Γ determines the whole spectrum. We give examples to show that this is not the case if the lattice has torsion.