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

TL;DR

This paper establishes bounds on the rank of the Hermitian adjacency matrix of complex unit gain graphs in terms of the graph's matching and cyclomatic numbers, extending known results from simpler graph classes.

Contribution

It generalizes existing bounds for undirected, mixed, and signed graphs to complex unit gain graphs and characterizes extremal cases.

Findings

Bounds on the rank in terms of matching and cyclomatic numbers.

Characterization of graphs achieving extremal rank bounds.

Extension of known graph spectral results to complex gain graphs.

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 $\mathbb{T}= \{ z \in C:|z|=1 \}$ 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 graphs $(G, \mathbb{T}, \varphi)$ 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 graphs. Moreover, we show that $2m(G-V_{0}) \leq r(G, \varphi) \leq 2m(G)+b(G)$ 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 $S \subset V(G)$.