A simple proof of the spectral excess theorem for distance-regular graphs

Abstract

The spectral excess theorem provides a quasi-spectral characterization for a (regular) graph Γ with d + 1 distinct eigenvalues to be distance-regular graph, in terms of the excess (number of vertices at distance d ) of each of its vertices. The original approach, due to Fiol and Garriga in 1997, was obtained by using a local approach, so giving a characterization of the so-called pseudo-distance-regularity around a vertex. In this paper we present a new simple projection method based in a global point of view, and where the mean excess plays an essential role.