$ T $ is a rooted tree with some vertex as its root.

[@Diestel Proposition 1.5.6] \[Theo.1\] Let $ G $ be a connected graph. Then $ G $ has a normal spanning tree with any specified vertex as its root.

Let $ G $ be a connected graph with spanning tree $ T $. Then for each edge $ e \in  E(G)\setminus E(T) $, adding the edge $e$ to $T$ creates a unique cycle in $T \cup \{e\}$. This cycle is called a *fundamental cycle* of $ G $. Let $ \overrightarrow{G} $ be an oriented graph after assigning some direction for each edge of $ G $. Suppose $
    (\overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}}) $ is an ordering of the directed edges of $ \overrightarrow{G} $. The *incidence matrix* of $ \overrightarrow{G} $ is an $ m \times n $ matrix whose rows and columns are indexed by vertices and edges of $ \overrightarrow{G} $, respectively, is denoted by $ Q(\overrightarrow{G}) $, and is defined as follows: $$(i,j)th\text{ entry of $ Q(\overrightarrow{G})$}=q_{ij}=\left\{
    \begin{array}{rl}
    0 & \text{if the vertex $ v_{i} $ and the edge $ \overrightarrow{e_{j}}$ are not incident, } \\
    1 & \text{if the edge $ \overrightarrow{e_{j}} $ 