Mixed graphs with cut vertices having exactly two positive eigenvalues

TL;DR

This paper characterizes mixed graphs with cut vertices that have exactly two positive eigenvalues in their Hermitian adjacency matrix, expanding understanding of spectral properties of mixed graphs.

Contribution

It provides a complete characterization of mixed graphs with cut vertices and positive inertia index 2, a novel spectral property analysis.

Findings

Characterization of mixed graphs with positive inertia index 2

Extension of previous work on graphs with one positive eigenvalue

Insights into spectral properties related to cut vertices

Abstract

A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was introduced by Liu and Li, independently by Guo and Mohar, in the study of graph energy. Recently, Yuan et al. characterized the mixed graphs with exactly one positive eigenvalue. In this paper, we study the positive inertia indices of mixed graphs and characterize the mixed graphs with cut vertices having positive inertia index 2.