On the inertia index of a mixed graph with the matching number

TL;DR

This paper investigates the positive and negative inertia indices of mixed unicyclic graphs, providing bounds and characterizations related to the eigenvalues of their Hermitian adjacency matrices.

Contribution

It introduces bounds and characterizations for the inertia indices of mixed graphs, focusing on unicyclic structures and their eigenvalue properties.

Findings

Established bounds for inertia indices of mixed graphs.

Characterized graphs attaining extremal inertia bounds.

Analyzed the inertia indices specifically for mixed unicyclic graphs.

Abstract

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of $G$, where $G$ is the underlying graph of $\widetilde{G}$. The positive inertia index, denoted by $p^{+}(G)$, and the negative inertia index, denoted by $n^{-}(G)$, of a mixed graph $\widetilde{G}$ are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of $\widetilde{G}$, respectively. In this paper, we study the positive and negative inertia index of the mixed unicyclic graph. Moreover, we give the upper and lower bounds of the positive and negative inertia index of the mixed graph, and characterize the mixed graphs which attain the upper and lower bounds respectively.