The row left rank of quaternion unit gain graphs in terms of pendant vertices

Abstract

Let $\tilde{G}=(G,U(\mathbb{Q}),\phi )$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\tilde{G}$, $U(\mathbb{Q})=\{q\in \mathbb{Q}:|q|=1\}$ and $\phi :\overrightarrow{E}\to U(\mathbb{Q})$ is the gain function such that $\phi (e_{ij})=\phi (e_{ji}{)}^{-1}=\overline{\phi (e_{ji})}$ for any adjacent vertices $v_i$ and $v_j$. Let $A(\tilde{G})$ be the adjacency matrix of $\tilde{G}$ and let $r(\tilde{G})$ be the row left rank of $\tilde{G}$. In this paper, we prove some lower bounds on the row left rank of $U(\mathbb{Q})$-gain graphs in terms of pendant vertices. All corresponding extremal graphs are characterized.