Several Adjacency Matrices For Digraphs and Their Spectra

Several results on spectra of digraphs related to structural properties of directed graphs will be presented.the question of which kind of adjacency matrix to use for digraphs seems even more important thanfor undirected graphs, since each such kind of matrix has its pros and cons. we will start with the usualadjacency matrix and relate it to some (older) results on digraphs with walk-regular properties. wenext consider the recently becoming quite popular hermitian adjacency matrix, which has the advantageof having real eigenvalues and present some results on spectral characterizations. finally, we consider avariation of the hermitian adjacency matrix (introduced recently by mohar) that we call the eisensteinmatrix, and that can be used even for signed digraphs.