connected graph with $n$ vertices and $m$ edges. By Theorem \[Theo.1\], $G$ contains a normal spanning tree $ T $ with specified vertex $ r $ as its root. Let $ \overrightarrow{C_{1}(T)}, \overrightarrow{C_{2}(T)}, \dots,$ $ \overrightarrow{C_{m-n+1}(T)} $ be the directed fundamental cycles in $ \overrightarrow{G_{T}} $. For $1 \leq k \leq m$, let $ \overrightarrow{e_{k}} $ denote the directed edges of $\overrightarrow{G_{T}} $, and let $( \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}})$ be an ordering of the directed edges of $ \overrightarrow{G_{T}} $. Let $Q$ be the incidence matrix of $ \overrightarrow{G_{T}} $ associated with $( \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}})$. For any cycle $ C $ in $ G $, let $ I(\overrightarrow{C}) $ denotes the incidence vector of a directed cycle $ \overrightarrow{C}$ with respect to $Q$. Then $ I(\overrightarrow{C}) $ is in the null space of $ Q $. Since the set of all incidence vectors of the directed fundamental cycles in $ \overrightarrow{G_{T}} $ forms a basis for the cycle subspace of $ G $, $ I(\overrightarrow{C}) $ can be expressed as $I(\overrightarrow{C})=c_{1}I(\overrightarrow{C_{1}(T)})+c_{2}I(\overrightarrow{C_{2}(T)})+ \dots+ c_{m-n+1}I(\overrightarrow{C_{m-n+1}(T)}) $, for some real numbers $c_1, c_2, \dots, c_{m-n+1}$.

The gain of any directed cycle $ \overrightarrow{C} $ in $ \Phi $ is $ 