Spectral Fundamentals and Characterizations of Signed Directed Graphs

Abstract: The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts. To represent such signed directed graphs, we use a striking equivalence to $\mathbb{T}_6$-gain graphs to formulate a Hermitian adjacency matrix, whose entries are the unit Eisenstein integers $\exp(k\pi i/3),$ $k\in \mathbb{Z}_6.$ Many well-known results, such as (gain) switching and eigenvalue interlacing, naturally carry over to this paradigm. We show that non-empty signed directed graphs whose spectra occur uniquely, up to isomorphism, do not exist, but we provide several infinite families whose spectra occur uniquely up to switching equivalence. Intermediate results include a classification of all signed digraphs with rank $2,3$, and a deep discussion of signed digraphs with extremely few (1 or 2) non-negative (eq. non-positive) eigenvalues.