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

Abstract

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