hence its eigenvalues are real. When $\varphi(\overrightarrow{e_{st}})=1$ for all $\overrightarrow{e_{st}}$, then $A(\Phi)=A(G)$. Thus we can consider $G$ as a $\mathbb{T}$-gain graph and we write this by $(G,1)$. For a square matrix $ B $ with complex entries, $\sigma(B)$ and $\rho(B)$ denote the spectrum and the spectral radius of $ B $, respectively.

A cycle is called [*directed cycle*]{} if its all edges are oriented in same direction. A directed cycle obtained from a cycle $C$ is denoted by $ \overrightarrow{C} $. For any cycle, there are only two directed cycles associated with it. Let $ \overrightarrow{C} \equiv v_{1} \overrightarrow{e_{12}}v_{2}\overrightarrow{e_{23}} \dots v_{k}\overrightarrow{e_{k1}}v_{1} $ be a directed cycle in a $ \mathbb{T}$-gain graph $ \Phi=(G, \varphi) $, then the gain of this cycle, denoted by $\varphi(\overrightarrow{C})$, is defined as $\varphi(\overrightarrow{C}) := \varphi(\overrightarrow{e_{12}})\varphi(\overrightarrow{e_{23}}) \cdots \varphi(\overrightarrow{e_{k1}}) $. If $\varphi(\overrightarrow{C})$ =1, we call the underlying cycle $ C $ as neutral in $ \Phi $. A $\mathbb{T} $-gain graph $ \Phi $ is said to be *balanced* if all the cycles in $ G $ are neutral in $ \Phi $.

Any function from the vertex set of $G$ to the complex unit circle $\mathbb{T}$ is called a *switching function*. Two $\mathbb{T}$-gain graphs $\Phi_1=(G,\varphi_1)$ and 