Spectra of Regular Polytopes

Introduction As is well known, the combinatorial problem of counting paths of length n between two xed vertices in a graph reduces to raising the adjacency matrix A of the graph to the n-th power ((B], p. 11). For an undirected graph, A is symmetric and the problem above simpliies considerably if its spectrum (A) is known and contains few distinct elements. Spectra of graphs, meaning spectra of the corresponding adjacency matrices, have been an active subject for decades ((CDS]). In this paper, we compute the spectra with multiplicities of the adjacency graphs of all regular polytopes in R n , or, for brevity, spectra of polytopes. We begin with a short theoretical setup and then proceed to the computations.