The left row rank of quaternion unit gain graphs in terms of girth

TL;DR

This paper establishes a lower bound on the left row rank of quaternion unit gain graphs based on girth and characterizes graphs achieving specific rank values, extending known results from simpler graph types.

Contribution

It introduces a lower bound for the left row rank in quaternion unit gain graphs and characterizes graphs with particular rank values, generalizing previous results from simpler graph classes.

Findings

Proves that the left row rank is at least girth minus two.

Characterizes graphs with rank equal to girth, girth minus one, and girth minus two.

Identifies all quaternion unit gain graphs with rank 2.

Abstract

Let $\Phi=(G,U(\mathbb{Q}),\varphi)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph). The adjacency matrix of $\Phi$ is denoted by $A(\Phi)$ and the left row rank of $\Phi$ is denoted by $r(\Phi)$. If $\Phi$ has at least one cycle, then the length of the shortest cycle in $\Phi$ is the girth of $\Phi$, denoted by $g$. In this paper, we prove that $r(\Phi)\geq g-2$ for $\Phi$. Moreover, we characterize $U(\mathbb{Q})$-gain graphs satisfy $r(\Phi)=g-i$ ($i=0,1,2$) and all quaternion unit gain graphs with rank 2. The results will generalize the corresponding results of simple graphs (Zhou et al. Linear Algebra Appl. (2021), Duan et al. Linear Algebra Appl. (2024) and Duan, Discrete Math. (2024)), signed graphs (Wu et al. Linear Algebra Appl. (2022)), and complex unit gain graphs (Khan, Linear Algebra Appl. (2024)).