\[def.1\], it is clear that all the cycles $ C_{1}(T), C_{2}(T), \dots , C_{m-n+1}(T) $ in $ G $ are the directed cycles in $ \overrightarrow{G_{T}} $. We denote them by $ \overrightarrow{C_{1}(T)}, \overrightarrow{C_{2}(T)},\dots, \overrightarrow{C_{m-n+1}(T)} $, and call them as *directed fundamental cycles* in $ \overrightarrow{G_{T}} $.

For any $ \mathbb{T} $-gain graph $ \Phi=(G, \varphi) $, if $ \varphi(\overrightarrow{e_{k}}) =e^{i\theta_{k}}$ for all $ 1 \leq k \leq m$ and $( \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}})$ is an ordering of the directed edges of $ \overrightarrow{G_{T}} $, then we can associate a unique $m$-vector $ \theta(\Phi):= (\theta_{1}, \theta_{2}, \dots, \theta_{m})$ with $ \Phi $.

            

\[Theo3.1\] Let $ \varphi_{1} $ and $  \varphi_{2} $ be two $ \mathbb{T} $-gains on a connected graph $ G $ with $ n $ vertices and $ m $ edges. Let $ T $ be a normal spanning tree of $ G $ and $ \overrightarrow{C_{j}(T)} $, $ j=1,2, \dots, (m-n+1) $ be the directed fundamental cycles in $ \overrightarrow{G_{T}} $. Then $ \varphi_{1}(\overrightarrow{C_{j}(T)} )= \varphi_{2}(\overrightarrow{C_{j}(T)} ) $, for all $ j=1,2, \dots (m-n+1) $ if and only if $ \varphi_{1}(\overrightarrow{C})=\varphi_{2}(\overrightarrow{C}) $, for any cycle $ C $ in $ G $.

Let $G$ be a 