A SPECTRAL EXCESS THEOREM FOR DIGRAPHS WITH NORMAL LAPLACIAN MATRICES

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 authorshave given some variations of this theorem. motivated by this, we give the corresponding resultby using the laplacian spectrum for digraphs. we also illustrate this laplacian spectral excesstheorem for digraphs with few laplacian eigenvalues and we show that any strongly connectedand 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.