On a version of the spectral excess theorem.

Given a regular (connected) graph Γ = (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 Γ 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. [ABSTRACT FROM AUTHOR]