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

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