The rank of a complex unit gain graph in terms of the rank of its underlying graph

TL;DR

This paper establishes bounds for the rank of complex unit gain graphs based on their underlying graphs' ranks and cycle space dimensions, providing insights into their spectral properties and extremal cases.

Contribution

It introduces bounds for the rank of complex unit gain graphs in terms of the underlying graph's rank and cycle space dimension, characterizing extremal cases.

Findings

Bounds for $r( ext{ extPhi})$ in terms of $r(G)$ and $ heta(G)$

Ratio bounds for $r( ext{ extPhi})/r(G)$ involving $ heta(G)$

Characterization of extremal graphs achieving bounds

Abstract

Let $\Phi=(G, \varphi)$ be a complex unit gain graph (or $\mathbb{T}$-gain graph) and $A(\Phi)$ be its adjacency matrix, where $G$ is called the underlying graph of $\Phi$. The rank of $\Phi$, denoted by $r(\Phi)$, is the rank of $A(\Phi)$. Denote by $\theta(G)=|E(G)|-|V(G)|+\omega(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\omega(G)$ are the number of edges, the number of vertices and the number of connected components of $G$, respectively. In this paper, we investigate bounds for $r(\Phi)$ in terms of $r(G)$, that is, $r(G)-2\theta(G)\leq r(\Phi)\leq r(G)+2\theta(G)$, where $r(G)$ is the rank of $G$. As an application, we also prove that $1-\theta(G)\leq\frac{r(\Phi)}{r(G)}\leq1+\theta(G)$. All corresponding extremal graphs are characterized.