Mixed graphs with smallest eigenvalue greater than $-\frac{\sqrt{5}+1}{2}$

TL;DR

This paper characterizes mixed graphs with smallest Hermitian eigenvalue greater than -rac{\u221a5+1}{2}, identifying three infinite classes and 30 scattered graphs, and introduces a new switching equivalence class.

Contribution

It provides a complete characterization of mixed graphs with a specific eigenvalue bound, including new classes and a switching equivalence result.

Findings

Identified three infinite classes of mixed graphs meeting the eigenvalue condition.

Found 30 scattered mixed graphs with the specified eigenvalue property.

Discovered a new class of mixed graphs switching equivalent to their underlying graphs.

Abstract

The classical problem of characterizing the graphs with bounded eigenvalues may date back to the work of Smith in 1970. Especially, the research on graphs with smallest eigenvalues not less than $-2$ has attracted widespread attention. Mixed graphs are natural generalization of undirected graphs. In this paper, we completely characterize the mixed graphs with smallest Hermitian eigenvalue greater than $-\frac{\sqrt{5}+1}{2}$, which consists of three infinite classes of mixed graphs and $30$ scattered mixed graphs. By the way, we get a new class of mixed graphs switching equivalent to their underlying graphs.