The rank of a complex unit gain graph in terms of the matching number

Abstract

A complex unit gain graph (or $\mathbb{T}$-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 $\mathbb{T}=\{z\in C:|z|=1\}$ and a gain function $\phi :\overrightarrow{E}\to \mathbb{T}$ with the property that $\phi (e_{i,j})=\phi (e_{j,i}{)}^{-1}$. In this paper, we prove that $2m(G)-2c(G)\le r(G,\phi )\le 2m(G)+c(G)$, where $r(G,\phi )$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G,\phi )$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, the complex unit gain graphs $(G,\mathbb{T},\phi )$ with $r(G,\phi )=2m(G)-2c(G)$ and $r(G,\phi )=2m(G)+c(G)$ are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graphs. Moreover, we show that $2m(G-V_0)\le r(G,\phi )\le 2m(G)+b(G)$ holds for any subset $V_0$ of $V(G)$ such that $G-V_0$ is acyclic and $b(G)$ is the minimum integer $|S|$ such that $G-S$ is bipartite for $S\subset V(G)$.