graph $ G $, each undirected edge $ e_{st} \in E(G) $ is associated with a pair of oriented edges, namely $ \overrightarrow{e_{st}} $ and $ \overrightarrow{e_{ts}} $. Set of all such oriented edges of a simple graph $ G $ is known as *oriented edge set* of $ G $, and is denoted by $ \overrightarrow{E}(G) $. A *$\mathbb{T}$-gain graph (or complex unit gain graph)* on a simple graph $ G $ is a triple $\Phi=(G,\mathbb{T},\varphi)$ such that the map (*the gain function*) $\varphi:\overrightarrow{E}(G)\rightarrow\mathbb{T}$ satisfies $\varphi(\overrightarrow{e_{st}})=\varphi(\overrightarrow{e_{ts}})^{-1}$. That is, for an oriented edge $\overrightarrow{e_{st}}$, if we assign a value $g$ (the *gain* of the edge $ \overrightarrow{e_{st}}$) from $\mathbb{T}$, then assign $g^{-1 }$ to the oriented edge $\overrightarrow{e_{ts}}$. For simplicity, we use $\Phi=(G,\varphi)$ to denote a $\mathbb{T}$-gain graph instead of $\Phi=(G,\mathbb{T},\varphi)$. We call $ \varphi $ as a $ \mathbb{T}$-gain on $ G $ if $ \Phi=(G, \varphi) $ is a $ \mathbb{T} $-gain graph on $ G $. In [@reff1], the author studied the notion of the adjacency matrix $A(\Phi)=(a_{st})_{n\times n}$ of a $\mathbb{T}$-gain graph $\Phi$. The entries of $A(\Phi)$ are given by $$a_{st}=\begin{cases}
    \varphi(\overrightarrow{e_{st}})&\text{if } \mbox{$v_s\sim v_t$},\\
    0&\text{otherwise.}\end{cases}$$ It is clear that the matrix $A(\Phi)$ is Hermitian, and 