Spectral fundamentals and characterizations of signed directed graphs

The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge graph, have received substantially less attention than those their undirected and/or unsigned counterparts. To represent such we use striking equivalence T6-gain graphs formulate Hermitian adjacency matrix, whose entries are the unit Eisenstein integers exp⁡(kπi/3), k∈Z6. Many well-known results, as (gain) switching and eigenvalue interlacing, carry over this paradigm. We show that non-empty spectra occur uniquely, up isomorphism, do not exist, but provide several infinite families uniquely equivalence. Intermediate results include classification all digraphs with rank 2,3, deep discussion extremely few (1 or 2) non-negative (eq. non-positive) eigenvalues.