A spectral excess theorem for digraphs with normal Laplacian matrices

The spectral excess theoremý, ýdue to Fiol and Garriga in 1997ý, ýis an important resultý, ýbecause it gives a good characterizationý ýof distance-regularity in graphsý. ýUp to nowý, ýsome authors have given some variations of this theoremý. ýMotivated by thisý, ýwe give the corresponding result by using the Laplacian spectrum for digraphsý. ýWe 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 eigenvaluesý, ýis distance-regularý. ýHence such a digraph is strongly regular with girth g=2 or g=3ý.