On a version of the spectral excess theorem - Miquel Àngel Fiol; Safet Penjic (Electronic Journal of Graph Theory and Applications (EJGTA), 2020)
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 AD 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.