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 $\Phi =(G,\mathbb{T},\phi )$ ($(G,\phi )$ for short) consisting of a graph $G$ as the underlying graph of $(G,\phi )$, $\mathbb{T}=\{z\in C:|z|=1\}$ is a subgroup of the multiplicative group of all nonzero complex numbers ${\mathbb{C}}^{\times }$ and a gain function $\phi :\overrightarrow{E}\to \mathbb{T}$ such that $\phi (e_{ij})=\phi (e_{ji}{)}^{-1}=\overline{\phi (e_{ji})}$. In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph $(G,\phi )$ with order $n$, and prove that $2n-2c(G)\le r(G,\phi )+2\alpha (G)\le 2n$. Where $r(G,\phi )$, $\alpha (G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $A(G,\phi )$, 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.

arXiv admin note: substantial text overlap with arXiv:1907.07837, arXiv:1909.07555