  The spectral excess theorem states that, in a regular graph G, the average
excess, which is the mean of the numbers of vertices at maximum distance from a
vertex, is bounded above by the spectral excess (a number that is computed by
using the adjacency spectrum of G), and G is distance-regular if and only if
equality holds. In this note we prove the corresponding result by using the
Laplacian spectrum without requiring regularity of G.
