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## Main Question or Discussion Point

I'm curious as to why the following theorem always works with a graph of vertices and the number of steps between the vertices when placed into an adjacency matrix.

If anyone can provide any insight or explanation, I would appreciate it.

The theorem makes sense to me, but I'm just unsure mathematically why exactly it works the way it does.

Thanks!

*If A is the adjacency matrix of a graph G (with vertices v1,…, vn), the (i, j)-entry of Ar represents the number of distinct r-walks from vertex vi to vertex vj in the graph.*If anyone can provide any insight or explanation, I would appreciate it.

The theorem makes sense to me, but I'm just unsure mathematically why exactly it works the way it does.

Thanks!