On a version of the spectral excess theorem

Miquel Àngel Fiol (Departament de Matem\`atiques, Universitat Polit\' ecnica de Catalunya, Barcelona Graduate School of Mathematics, Catalonia, Spain)
Safet Penjic (Andrej Maru\v{s}i\v{c} Institute, University of Primorska, Muzejski trg 2 6000 Koper, Slovenia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 8, No 2 (2020): Electronic Journal of Graph Theory and Applications

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.