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

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)).