Adjacency matrix of a directed graph ($A$)

The adjacency matrix of a finite directed graph has entry $A_{ij}$ equal to the number of directed edges from vertex $i$ to vertex $j$. The entry $(A^n{)}_{ij}$ counts length-$n$ directed walks from $i$ to $j$, so $\mathrm{tr}(A^n)$ counts pointed closed directed walks of length $n$.