Adjacency matrix of a graph ($A(G)$)

The adjacency matrix of a finite simple graph has entry $A_{uv}=1$ when vertices $u,v$ are adjacent and zero otherwise.

For every nonnegative integer $k$, the entry $(A^k{)}_{uv}$ equals the number of length-$k$ walks from $u$ to $v$. This follows by induction, since matrix multiplication appends one adjacent vertex to each walk.