On symmetric spectra of Hermitian adjacency matrices for non-bipartite mixed graphs

TL;DR

This paper explores the relationship between bipartiteness and spectral symmetry in mixed graphs with $ heta$-Hermitian adjacency matrices, revealing conditions under which symmetry implies bipartiteness and constructing non-bipartite graphs with symmetric spectra.

Contribution

It establishes the equivalence between bipartiteness and spectral symmetry for certain algebraic angles and provides explicit constructions of non-bipartite graphs with symmetric spectra for rational multiples of $\pi$.

Findings

Spectral symmetry implies bipartiteness when $ heta$ is algebraic.

The equivalence breaks down for rational multiples of $\pi$.

Constructed non-bipartite graphs with symmetric spectra for $ heta \

Abstract

We study the equivalence between bipartiteness and symmetry of spectra of mixed graphs, for $\theta$-Hermitian adjacency matrices defined by an angle $\theta \in (0, \pi]$. We show that this equivalence holds when, for example, an angle $\theta$ is an algebraic number, while it breaks down for any angle $\theta \in \mathbb{Q}\pi$. Furthermore, we construct a family of non-bipartite mixed graphs having the symmetric spectra for given $\theta \in \mathbb{Q}\pi$.