On a version of the spectral excess theorem

Abstract

Given a regular (connected) graph G =( X,E ) with adjacency matrix A , d +1 distinct eigenvalues, and diameter D , we give a characterization of when its distance matrix A D is a polynomial in A , in terms of the adjacency spectrum of G and the arithmetic (or harmonic) mean of the numbers of vertices at distance at most D -1 from every vertex. The same result is proved for any graph by using its Laplacian matrix L and corresponding spectrum. When D = d we reobtain the spectral excess theorem characterizing distance-regular graphs.