The AdjacencyMatrix command returns the adjacency matrix of a graph G whose rows and columns are indexed by the vertices. The entry $i,j$ of this matrix is 1 if there is an edge from vertex i to vertex j and 0 otherwise.

The default output is an n by n Matrix with the following properties:

If G is directed or undirected: datatype=anything and order=C_order

If G is undirected: shape=symmetric, storage=triangular[upper],

If G is directed: storage=rectangular, shape=[]

If G is sparse, i.e., |E| << |V|^2 then storage=sparse will be used.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\&colon;$

$G≔\mathrm{Graph}\left(\left[1\&comma;2\&comma;3\&comma;4\right]\&comma;\mathrm{Trail}\left(1\&comma;2\&comma;3\&comma;4\&comma;1\right)\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 4\; edge(s)}$

$\mathrm{AdjacencyMatrix}\left(G\right)$

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

$\mathrm{Neighbors}\left(G\right)$

$\left[\left[2\&comma;4\right]\&comma;\left[1\&comma;3\right]\&comma;\left[2\&comma;4\right]\&comma;\left[1\&comma;3\right]\right]$

$H≔\mathrm{Digraph}\left(\left[1\&comma;2\&comma;3\&comma;4\right]\&comma;\mathrm{Trail}\left(1\&comma;2\&comma;3\&comma;4\&comma;1\right)\right)$

$H≔\mathrm{Graph\; 2:\; a\; directed\; unweighted\; graph\; with\; 4\; vertices\; and\; 4\; arc(s)}$

$\mathrm{AdjacencyMatrix}\left(H\right)$

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

$\mathrm{Departures}\left(H\right)$

$\left[\left[2\right]\&comma;\left[3\right]\&comma;\left[4\right]\&comma;\left[1\right]\right]$