\[Theo3.1\], the choice of normal spanning tree is immaterial. The next corollary proves that the gains of the directed fundamental cycles determine whether two $ \mathbb{T} $-gain graphs on a same underlying graph are switching equivalent or not.

\[cor.3.2\] Let $ \varphi_{1} $ and $ \varphi_{2} $ be two $ \mathbb{T} $-gains on a connected graph $ G $ with $ n $ vertices and $ m $ edges, and $T$ be any normal spanning tree of $ G $. Then $ \Phi_{1} \sim \Phi_{2} $ if and only if $ \varphi_{1}( \overrightarrow{C_{k}(T)})=\varphi_{2}( \overrightarrow{C_{k}(T)}) $, $k=1,2,\dots, (m-n+1) $.

If $ \Phi_{1} \sim \Phi_{2} $, then it is clear that $ \varphi_{1}( \overrightarrow{C_{k}(T)})=\varphi_{2}( \overrightarrow{C_{k}(T)}) $, $ k=1,2,\dots, (m-n+1) $. Proof of the converse follows from Theorem \[Theo3.1\] and Theorem \[Theo.3\].

Here we are giving an algorithm to check whether two $ \mathbb{T} $-gains are switching equivalent or not.

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  [**Algorithm 1.**]{} Identify whether $ \Phi_{1} \sim \Phi_{2} $ or not.
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  1: Compute a normal spanning $T $.

  2: Compute $( \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots,
       \overrightarrow{e_{m}})$, the edge ordering with respect to $ T $.

  3: Two gains $ \theta(\Phi_{1}) =(\theta_{1}, \theta_{2}, \dots, \theta_{m})$ and $ \theta(\Phi_{2}) =(\psi_{1}, \psi_{2}, \dots, \psi_{m})$.

  4: 