Spectral Fundamentals and Characterizations of Signed Directed Graphs

TL;DR

This paper explores the spectral properties of signed directed graphs using a novel Hermitian adjacency matrix based on Eisenstein integers, revealing unique spectral characteristics and classifying graphs by rank and eigenvalue distribution.

Contribution

It introduces a new spectral framework for signed directed graphs via $ ext{T}_6$-gain graphs and classifies graphs with specific spectral properties, advancing understanding of their structure.

Findings

Spectral properties are characterized using a Hermitian adjacency matrix with Eisenstein integers.

Non-empty signed directed graphs with unique spectra up to isomorphism do not exist.

Classified all signed digraphs with rank 2 and 3, and analyzed graphs with few non-negative eigenvalues.

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.