A new approach to the spectral excess theorem for distance-regular graphs

The Spectral Excess Theorem provides a quasi-spectral characterization for a (regular) graph $\Gamma$ with $d+1$ different eigenvalues to be distance-regular graph, in terms of the mean (d-1)-excess of its vertices.\ The original approach, due to Fiol and Garriga in $1997$, was obtained in a wide context from a local point of view, so giving a characterization of the so-called pseudo-distance-regularity around a vertex.\ In this paper we present a new simple method based in a global point of view, and where the mean degree of the distance-$d$ graph $\Gamma_d$ plays an essential role