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

TL;DR

This paper establishes bounds on the inertia indices of complex unit gain graphs in terms of matching and cyclomatic numbers, providing characterizations for extremal cases.

Contribution

It introduces bounds for positive and negative eigenvalues of complex unit gain graphs based on graph invariants, with characterizations of extremal graphs.

Findings

Bounds for positive and negative eigenvalues in terms of matching and cyclomatic numbers.

Characterizations of graphs attaining the bounds.

Extension of spectral graph theory to complex unit gain graphs.

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.