Finding the adjacency matrix for any given quiver and some collection of words.

For a directed graph (quiver) $Q$ with $n$ vertices and without multiple arrows, we have the adjacency matrix $A$, in which $A(i,j)=1$, if there is an arrow from $i$ to $j$, and $0$ elsewhere. This matrix encodes the graph in a useful way. (This argument also works for undirected graphs) Then we could use the $k$-th power of the adjacency matrix to count the number of paths of length $k$ between any pairs of vertices. Equivalently, we could read the entry $A^k(i,j)$ to count the number of words of length $k+1$, starting with letter $i$ and terminating with letter $j$.

Here are my questions:

• Thinking about the adjacency matrix, it occurred to me that we might be able to use the same idea for any quiver, including the ones with multiple arrows, such that the powers of the matrix are as useful as the adjacency matrix. I checked the idea and I don't see any problems and everything goes smoothly. I was wondering if you could insure me that I am not missing something here!

• It seems clear that for a given collection of words of the same length $k$, we may not necessarily have an adjacency matrix (and consequently a quiver) whose $k$-th power could be translated into the given collection of words. But, do we know the sufficient conditions for such a collection, under which we could uniquely find the adjacency matrix and then the quiver?