The Hoffman program for mixed graphs

TL;DR

This paper investigates the spectral properties of mixed graphs, specifically identifying graphs with spectral radii below a certain threshold and characterizing their limit points, extending Hoffman's program.

Contribution

It determines all mixed graphs without negative 4-cycles with spectral radius at most rac{rac{2+rac{5}{2}}}{} and identifies all limit points of spectral radii.

Findings

Characterization of mixed graphs without negative 4-cycles with spectral radius rac{rac{2+rac{5}{2}}}{}

Identification of all limit points of spectral radii of mixed graphs

Extension of Hoffman's program to mixed graphs

Abstract

We consider Hoffman's program about the limit points of the spectral radius of the Hermitian adjacency matrix of mixed graphs. In particular, we determine all mixed graphs without negative $4$-cycle whose spectral radius does not exceed $\sqrt{2+\sqrt{5}}$, and identify all limit points of spectral radii of mixed graphs.