...umber is given by [1H1 : Res G H1 ρλG], where 1H1 is the 5 trivial representation on H1. Similarly, the dimension of the λ-eigenspace on Γ2 is equal to [1H2 : Res G H2 ρλG]. By Frobenius reciprocity (=-=[4]-=-), we have [1Hi : Res G Hi ρλG] = [Ind G Hi 1Hi : ρ λ G] (17) for i = 1, 2. By our earlier remark, Ind GHi1Hi is just our quasi-regular representation piGHi , so we conclude that( dimension of λ-eigen...

...ry condition and one using the conjugacy-class condition – and consider the extent to which each proof is convertible. 5 Pesce’s proof and a converse The first proof is slightly adapted from a paper (=-=[2]-=-) by Hubert Pesce, and uses the language of representations. Proof For notational convenience, denote H1\Γ by Γ1 and H2\Γ by Γ2. We first observe that if λ is an eigenvalue of the adjacency operator o...

...s the adjacency spectrum of a graph and the numbers of closed n-walks (for n going from 0 to the number of vertices) determine one another. A similar result holds for closed nbt walks. Theorem 4 (See =-=[3]-=-.) Let Γ be a finite, undirected, regular graph. For each non-negative integer n, let Dn denote the total number of closed nbt n-walks in Γ. Then the adjacency spectrum of Γ determines the sequence D0...

...H2\Γ will then be isospectral, by the well-known Theorem 2 below. Robert Brooks raised the question whether all pairs of isospectral regular graphs arise in this way, and provided a partial answer in =-=[1]-=-. In this note, we examine two independent proofs of Theorem 2 and consider what they have to tell us about a possible converse. 2 Notation and definitions Let Γ be a finite, undirected graph with ver...