If $A$ is an adjacency matrix of a labeled multi-digraph is the $(i,j)^{th}$ coordinate of $A^n$ the number of directed walks from $i$ to $j$? I know this is true when $A$ is the adjacency matrix of a digraph, but what about for multi-digraphs i.e. directed graphs that can have loops and multiple arrows connecting adjacency vertices. Where just to review if $A$ is the adjacency matrix of a labeled multi-digraph $[A]_{i,j}$ is equal to the number of arrows with vertex tail $i$ and vertex head $j$. For instance the picture bellow shows an example of such an adjacency matrix for a multi-digraph:
