Editing Adjacency matrix (section)

===Matrix powers===
If {{mvar|A}} is the adjacency matrix of the directed or undirected graph {{mvar|G}}, then the matrix {{math|''A''<sup>''n''</sup>}} (i.e., the [[matrix multiplication|matrix product]] of {{mvar|n}} copies of {{mvar|A}}) has an interesting interpretation: the element {{math|{{nowrap|(''i'', ''j'')}}}} gives the number of (directed or undirected) [[Path (graph theory)|walks]] of length {{mvar|n}} from vertex {{mvar|i}} to vertex {{mvar|j}}. If {{mvar|n}} is the smallest nonnegative integer, such that for some {{mvar|i}}, {{mvar|j}}, the element {{math|{{nowrap|(''i'', ''j'')}}}} of {{math|''A''<sup>''n''</sup>}} is positive, then {{mvar|n}} is the distance between vertex {{mvar|i}} and vertex {{mvar|j}}. A great example of how this is useful is in counting the number of triangles in an undirected graph {{mvar|G}}, which is exactly the [[Trace (linear algebra)|trace]] of {{math|''A''<sup>3</sup>}} divided by 3 or 6 depending on whether the graph is directed or not. We divide by those values to compensate for the overcounting of each triangle. In an undirected graph, each triangle will be counted twice for all three nodes, because the path can be followed clockwise or counterclockwise : ijk or ikj. The adjacency matrix can be used to determine whether or not the graph is [[Connectivity (graph theory)|connected]].

If a directed graph has a [[nilpotent matrix|nilpotent]] adjacency matrix (i.e., if there exists {{mvar|n}} such that {{math|''A''<sup>''n''</sup>}} is the zero matrix), then it is a [[directed acyclic graph]].<ref>{{Cite journal|title=Matrices with Permanent Equal to One |last1=Nicholson |first1=Victor A |year=1975 |journal=Linear Algebra and Its Applications |issue=12 |page=187|url=https://core.ac.uk/download/pdf/82099476.pdf}}</ref>