n$ matrix defined as follows: $(s,t)th$ entry of the matrix is the gain of the edge starts from the vertex $v_{s}$ and ends at the vertex $v_{t}$, and is $0$ if there is no edge between the vertices $v_{s}$ and $v_{t}$. Particular cases of the notion of adjacency matrix of $\mathbb{T}$-gain graphs were considered with different gains in the literature [@Bap; @Kat]. In [@Kat], the authors considered complex weighted graphs with the weights are in $\{\pm1,\pm i\}$. In [@Ger], the authors studied some of the properties of the characteristic polynomial of the $\mathbb{T}$-gain graphs. For some interesting spectral properties of $\mathbb{T}$-gain graphs, we refer to [@reff1; @Reff2016; @Ger; @Zaslav; @Zas2]. A *directed graph(or digraph)* $ X $ is an order pair $ (V(X), E(X)) $, where $ V(X)=\{ v_{1}, v_{2}, \dots,v_{n}\} $ is the vertex set and $ E(X) $ is the directed edge set. A directed edge from the vertex $ v_{s} $ to the vertex $ v_{t} $ is denoted by $ \overrightarrow{e_{st}} $. If $ \overrightarrow{e_{st}} \in E(X)$ and $  \overrightarrow{e_{ts}}\in E(X)$, then the pair $ \{v_{s},v_{t}\} $ is called a *digon* of $ X $. The underlying graph of $ X $ is a simple undirected 