Abstract :
The spectral excess theorem, due to Fiol and Garriga in 1997, is an important result,
because it gives a good characterization of distance-regularity in graphs. Up to now, some authors
have given some variations of this theorem. Motivated by this, we give the corresponding result
by using the Laplacian spectrum for digraphs. We also illustrate this Laplacian spectral excess
theorem for digraphs with few Laplacian eigenvalues and we show that any strongly connected
and regular digraph that has normal Laplacian matrix with three distinct eigenvalues, is distance-
regular. Hence such a digraph is strongly regular with girth g = 2 or g = 3.
