Compute the incidence vectors $ I(\overrightarrow{C_{j}(T)}), j=1,2, \dots, (m-n+1) $, where $ \overrightarrow{C_{j}(T)} $ are

      the directed fundamental cycles with respect to $T$.

  5: Compute $ \varphi_{1}(\overrightarrow{C_{j}(T)})= e^{i \langle I(\overrightarrow{C_{j}(T)}),  \theta(\Phi_{1}) \rangle }$ and $ \varphi_{2}(\overrightarrow{C_{j}(T)})= e^{i \langle I(\overrightarrow{C_{j}(T)}),  \theta(\Phi_{2}) \rangle }$

      for all $ j=1,2, \dots, (m-n+1)$.

  6: Check $ \varphi_{1}(\overrightarrow{C_{j}(T)}) =\varphi_{2}(\overrightarrow{C_{j}(T)})$ for $ j=1,2, \dots (m-n+1) $.

  7: [**Return:** ]{} If step $(6)$ true, then $ \Phi_{1} \sim \Phi_{2} $; otherwise $ \Phi_{1} \nsim \Phi_{2} $.
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Proof of the following theorem is immediate from corollary \[cor.3.2\]. We give a proof of the same with out using corollary \[cor.3.2\].

\[Theo3.2\] 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 $ such that $\varphi_{1}(\overrightarrow{C_{j}(T)})=\varphi_{2}(\overrightarrow{C_{j}(T)}) $, for all $ j=1,2,\dots, (m-n+1) $. Then, $A(\Phi_1)$ and $A(\Phi_2)$ have the same spectrum. i.e., $\Phi_1$ and $\Phi_2$ are $\mathbb{T}$-cospectral.

By Theorem \[Theo3.1\], $ \varphi_{1}(\overrightarrow{C}) = \varphi_{2}(\overrightarrow{C}) $, for any cycle $ C $ in $ G $. There are exactly two directed cycles can be defined on an undirected cycle $ C $. 