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 $\\Phi=(G, {\\mathbb T}, \\varphi)$ (or $(G, \\varphi)$ for short) consisting of a simple graph $G$, as the underlying graph of $(G, \\varphi)$, the set of unit complex numbers $\\mathbb{T}= \\{ z \\in C:|z|=1 \\}$ % is a subgroup of the multiplicative group of all nonzero complex numbers $\\mathbb{C}^{\\times}$ and a gain function $\\varphi: \\overrightarrow{E} \\rightarrow \\mathbb{T}$ with the property that $\\varphi(e_{i,j})=\\varphi(e_{j,i})^{-1}$. In this paper, we prove that $2m(G)-2c(G) \\leq r(G, \\varphi) \\leq 2m(G)+c(G)$, where $r(G, \\varphi)$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G, \\varphi)$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, the complex unit gain graph $(G, \\mathbb{T}, \\varphi)$ with $r(G, \\varphi)=2m(G)-2c(G)$ and $r(G, \\varphi)=2m(G)+c(G)$ are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graph. Moreover, %the following upper bound and lower bound are obtained which is we show that $2m(G-V_{0}) \\leq r(G, \\varphi) \\leq 2m(G)+S$ holds for any $S\\subset V(G)$ such that $G-S$ is bipartite and any subset $V_0$ of $V(G)$ such that $G-V_0$ is acyclic.