$\Phi_2=(G,\varphi_2)$ are said to be *switching equivalent*, denoted by $\Phi_1\sim\Phi_2$, if there is a switching function $\zeta:V(G)\rightarrow\mathbb{T}$ such that $$\varphi_2(\overrightarrow{e_{st}})=\zeta(v_s)^{-1}\varphi_1(\overrightarrow{e_{st}})\zeta(v_t).$$

The switching equivalence of two $\mathbb{T}$-gain graphs can be defined in the following equivalent way: Two $\mathbb{T}$-gain graphs $\Phi_1=(G,\varphi_1)$ and $\Phi_2=(G,\varphi_2)$ are switching equivalent, if there exists a diagonal matrix $D_\zeta$ with diagonal entries from $\mathbb{T}$ such that $$A(\Phi_2)=D_\zeta^{-1}A(\Phi_1)D_\zeta.$$

[@Reff2016 Lemma 2.2]\[Theo.3\] Let $\varphi_{1}$ and $ \varphi_{2}$ be two $ \mathbb{T}$-gains on a connected graph $ G $. If for every cycle $ C $ in $ G $ there is a directed cycle $ \overrightarrow{C} $ such that $ \varphi_{1}(\overrightarrow{C})=\varphi_{2}(\overrightarrow{C}) $, then $ \Phi_{1} \sim \Phi_{2} $.

The following result gives an upper bound for the spectral radius of $A(\Phi)$ in terms of the maximum vertex degree $\Delta$ of $G$.

\[Theo.12\] Let $ \varphi $ be a $ \mathbb{T} $-gain on a graph $ G $. Then $ \rho(A(\Phi)) \leq \Delta. $

Next, we collect a couple of results related to the spectrum and the spectral radius of adjacency matrices of $\mathbb{T}$-gain graphs.

[@Our-gain1 Theorem 4.6] \[Theo.7\] Let $ \varphi $ be a $ \mathbb{T} $-gain on a connected graph $ G $. Then, $ \sigma(A(\Phi)) =\sigma(A(G))$ if and 