originates at the vertex $ v_{i},$  }\\
    -1 & \text{if the edge $ \overrightarrow{e_{j}} $ terminates at the vertex $ v_{i}.$  }
    \end{array}\right.$$ The null space of $  Q(\overrightarrow{G}) $ is known as the *cycle subspace* of $ G $. Let $ S $ be a subgraph of $ G $. The *incidence vector* of an oriented subgraph $ \overrightarrow{S} $ with respect to the ordering of the edges $ (\overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}}) $ is an $ m \times 1 $ vector defined as follows: If the edge $ e_{i} \in E(G)$ is in $ \overrightarrow{S} $ with its direction is same as that of $ \overrightarrow{e_{i}} $ (opposite to that of $ \overrightarrow{e_{i}} $, respectively) in $ \overrightarrow{G} $, then the $ i$th entry is $ 1$ ($-1$, respectively), and $ 0 $ otherwise. Then the set of incidence vectors of all fundamental cycles associated with $ T $ forms a basis for the cycle subspace of $ G $ [@Bapat Theorem 5.1].

In this article, we call $ \overrightarrow{e_{st}} $ as a directed edge in the context of digraphs, and we call the same as an oriented edge in the context of gain graphs. For any simple 