On mixed graphs whose Hermitian spectral radii are at most 2

TL;DR

This paper characterizes mixed graphs with Hermitian spectral radii at most 2, providing conditions for spectral radius decrease upon edge or vertex removal and classifying certain acyclic graphs.

Contribution

It offers a sufficient condition for decreasing spectral radius through deletions and characterizes specific mixed graphs with spectral radius at most 2 and no 4-cycle.

Findings

Hermitian spectral radius decreases when edges or vertices are removed under certain conditions

Characterization of mixed graphs with spectral radius at most 2 and no 4-cycle

Extension of previous classifications by Guo and Mohar

Abstract

A mixed graph is a graph with undirected and directed edges. Guo and Mohar in 2017 determined all mixed graphs whose Hermitian spectral radii are less than $2$. In this paper, we give a sufficient condition which can make Hermitian spectral radius of a connected mixed graph strictly decreasing when an edge or a vertex is deleted, and characterize all mixed graphs with Hermitian spectral radii at most $2$ and with no cycle of length $4$ in their underlying graphs.