The row left rank of a quaternion unit gain graph in terms of maximum degree

Abstract

Let $\Phi =(G,U(\mathbb{Q}),\phi )$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph) of order $n$, $A(\Phi )$ be the adjacency matrix of $\Phi$ and $r(\Phi )$ be the row left rank of $\Phi$. Let $\Delta$ be the maximum degree of $\Phi$. In this paper, we prove that $r(\Phi )\ge \frac{n}{\Delta }$. Moreover, if $\Phi$ is connected, we obtain that $r(\Phi )\ge \frac{n-2}{\Delta -1}$. All the corresponding extremal graphs are characterized.