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arXiv · 2108.01443

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

Yong Lu·Qi Wu
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Abstract

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

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Yong Lu, Qi Wu. 2021-08-01. Inertia indices of a complex unit gain graph in terms of matching number. https://arxiv.org/abs/2108.01443

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