Inertia indices of a complex unit gain graph in terms of matching number

Abstract

A complex unit gain graph is a triple $\varphi=(G, \mathbb{T}, \varphi)$ (or $G^{\varphi}$ for short) consisting of a simple graph $G$, as the underlying graph of $G^{\varphi}$, the set of unit complex numbers $\mathbb{T}={z\in \mathbb{C}: |z| = 1}$ and a gain function $\varphi: \overrightarrow{E}\rightarrow \mathbb{T}$ such that $\varphi(e_{i,j})=\varphi(e_{j,i}) ^{-1}$. Let $A(G^{\varphi})$ be adjacency matrix of $G^{\varphi}$. In this paper, we prove that $$m(G)-c(G)\leq p(G^{\varphi})\leq m(G)+c(G),$$ $$m(G)-c(G)\leq n(G^{\varphi})\leq m(G)+c(G),$$ where $p(G^{\varphi})$, $n(G^{\varphi})$, $m(G)$ and $c(G)$ are the number of positive eigenvalues of $A(G^{\varphi})$, the number of negative eigenvalues of $A(G^{\varphi})$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds, respectively.