The adjacency matrix is the n by n matrix (where n is the number of vertices in graph/digraph G) with rows and columns indexed by the vertices of G. Entry A_(u,v) is 1 if and only if {u,v} is an edge of G and 0 otherwise. It is easy to observe that if we just use a simple graph G, then its adjacency matrix must be symmetric, but if we use a digraph, then it is not necessarily symmetric.
i1 : D = digraph({{1,2},{2,3},{3,4},{4,3}},EntryMode=>"edges");
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i2 : adjacencyMatrix D
o2 = | 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
| 0 0 1 0 |
4 4
o2 : Matrix ZZ <--- ZZ
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i3 : G = graph({1,2,3,4}, {{1,2},{2,3},{3,4},{4,3}})
o3 = Graph{1 => {2} }
2 => {1, 3}
3 => {2, 4}
4 => {3}
o3 : Graph
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i4 : adjacencyMatrix G
o4 = | 0 1 0 0 |
| 1 0 1 0 |
| 0 1 0 1 |
| 0 0 1 0 |
4 4
o4 : Matrix ZZ <--- ZZ
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The object adjacencyMatrix is a method function.