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

TL;DR

This paper establishes bounds relating the rank, independence number, and cyclomatic number of complex unit gain graphs, providing characterizations for when the bounds are tight.

Contribution

It introduces bounds connecting the rank, independence number, and cyclomatic number of complex unit gain graphs and characterizes cases achieving the bounds.

Findings

Proves that 2n - 2c(G) ≤ r(G, φ) + 2α(G) ≤ 2n.

Characterizes properties of gain graphs reaching the lower bound.

Establishes a relationship among rank, independence number, and cyclomatic 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.