This is the directed cycle with $4$ vertices:
4 vertices">
This is the directed wheel with $5$ vertices in total:
5 vertices">
[I'm assuming the definition here. There might be other definitions, but the process is still the same.]
To write down an adjacency list or adjacency matrix, we need to give the vertices labels. The labels are arbitrary, but it's usually easier to choose labels such as $1,2,\ldots,n$. So let's label the vertices as follows:
4 labelled vertices">
and
5 labelled vertices">
An adjacency list is simply a list of the edges in the graph. We can just read these from the above drawings:
- The adjacency list for directed cycle with $4$ vertices has adjacency list is $(1,2),(2,3),(3,4),(4,1)$.
- The adjacency list for directed wheel with $5$ vertices has adjacency list is $(1,2),(2,3),(3,4),(4,1),(5,1),(5,2),(5,3),(5,4)$.
Note: these are directed edges, so e.g. $(5,1)$ is different to $(1,5)$.
The adjacency matrix of a directed graph $G$ is a $(0,1)$-matrix with a $1$ in cell $(i,j)$ whenever $(i,j)$ is an edge in $G$ (and $0$ otherwise).
- The adjacency matrix for directed cycle with $4$ vertices is
$\begin{array}{|cccc|} \hline 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ \hline \end{array}.$
- The adjacency matrix for directed wheel with $5$ vertices is
$\begin{array}{|ccccc|} \hline 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ \hline \end{array}.$
One caveat: the adjacency list and adjacency matrix will change depending on which way you label the vertices.