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

Abstract

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

17 pages. arXiv admin note: text overlap with arXiv:1909.07555 by other authors