Bounds for the rank of a complex unit gain graph in terms of the independence number

Abstract

A complex unit gain graph (or $\\mathbb{T}$-gain graph) is a triple $\\Phi=(G, \\mathbb{T}, \\varphi)$ ($(G, \\varphi)$ for short) consisting of a graph $G$ as the underlying graph of $(G, \\varphi)$, $\\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 $\\varphi: \\overrightarrow{E} \\rightarrow \\mathbb{T}$ such that $\\varphi(e_{ij})=\\varphi(e_{ji})^{-1}=\\overline{\\varphi(e_{ji})}$. In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph $(G, \\varphi)$ with order $n$, and prove that $2n-2c(G) \\leq r(G, \\varphi)+2\\alpha(G) \\leq 2n$. Where $r(G, \\varphi)$, $\\alpha(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $A(G, \\varphi)$, the independence number and the cyclomatic number of $G$, respectively. Furthermore, the properties of the complex unit gain graph that reaching the lower bound are characterized.