Graph

Graph consist of Nodes & Edges where m(row) is#edges & n(col) is#nodes

For the following example, the graph is consist of 4 nodes and 5 edges.

Incidence Matrices §

$$\begin{array}{ccccc}& nod{e}_1& nod{e}_2& nod{e}_3& nod{e}_4\\ edg{e}_1& -1& 1& 0& 0\\ edg{e}_2& 0& -1& 1& 0\\ edg{e}_3& -1& 0& 1& 0\\ edg{e}_4& -1& 0& 0& 1\\ edg{e}_5& 0& 0& -1& 1\end{array}$$

Write into Matrix form

$$A=\left[\begin{array}{cccc}-1& 1& 0& 0\\ 0& -1& 1& 0\\ -1& 0& 1& 0\\ -1& 0& 0& 1\\ 0& 0& -1& 1\end{array}\right]$$

Observation from the first three rows, we notice there is linear dependent relationship, which mean the three row spaces form an loop.