graph obtain from $ X $ by replacing a directed edge by an undirected edge and denoted by $ \Gamma(X) $. The *Hermitian adjacency matrix* [@Bojan; @Lie] of a digraph $ X $ is denoted by $H(X)$ and is defined as follows: $$\mbox{$ (s,t){th}$ entry of }H(X)=h_{st}=\begin{cases}
    1& \text{if both } \mbox{$\overrightarrow{e_{st}}$ \text{and} $\overrightarrow{e_{ts}} \in E(X)$},\\
    i& \text{if  } \mbox{$\overrightarrow{e_{st}} \in E(X)$ \text{and} $\overrightarrow{e_{ts}} \notin E(X)$},\\
    -i& \text{if  } \mbox{$\overrightarrow{e_{st}} \notin E(X)$ \text{and} $\overrightarrow{e_{ts}} \in E(X)$},\\
    0&\text{otherwise.}\end{cases}$$

The Hermitian adjacency matrix can be thought of as the adjacency matrix of a $\mathbb{T}$-gain graph with the gains are from $ \{1, \pm i\}$.

Given a graph $G$, one can associate uncountably many $\mathbb{T}$-gain graphs with $G$. Let $\mathbb{T}_{G} $ denote the collection of all $\mathbb{T}$-gain adjacency matrices on a graph $G$. First main objective of this article is to study the relationship between the spectral properties of various $\mathbb{T}$-gain adjacency matrices associated with $G$. In section \[co-spec\], we partition $\mathbb{T}_{G} $ using the gains of some special types of cycles(fundamental cycles) of $G$, and show that any two matrices inside the same class are cospectral\[Theorem \[Theo3.2\]\]. Also we observe that, either the specturm or the spectral radius of $\mathbb{T}$-gain adjacency matrices 