Graphing digraphs with the following vertex set

Question

Just want to make sure I did this correctly.. I think I did part a) correctly? Here is my solution for part a)

Not sure how to do b) and c) though. Any advice would be great. Thanks in advance

Answer 1

The trickiest part is the incidence matrix. You have a row for each vertex and a column for each edge. If $v$ is a vertex and $e$ is an edge, the $\langle v,e\rangle$ entry is $0$ if $e$ is not incident at $v$, $1$ if $e$ enters is $v$, $-1$ if $e$ leaves $v$ (I inadvertently reversed this in a comment that I’ve now erased), and (at least by the definition that I know) $2$ if $e$ is a loop at $v$. Thus, we get the following matrix:

$$\begin{bmatrix} 1&-1&0&-1&-1&1&0\\ -1&1&2&0&0&0&0\\ 0&0&0&0&1&0&1\\ 0&0&0&1&0&-1&-1 \end{bmatrix}$$