Exercise 3, § 1

Let $Q = (S,F, o, t)$ be a locally finite quiver. We define the matrix $M_Q= (m_{i,j})$, of type $(S,S)$ with entries in $\mathbf{Z}$, by

$m_{i,j}=$ Card$\{f\in F|o(f) =i$ and $t(f) =j\}$. The matrix $M_Q$ is called the adjacency matrix of Q.

a) Given quivers Q and $Q'$, give a necessary and sufficient condition on their adjacency matrices for the quivers Q and $Q'$ to be isomorphic.

b) Express in terms of the adjacency matrix $M_Q$ the number of paths of length $n$ in Q whose origin and term are given.

c) Exercise prove that the adjacency matrix of the quiver underlying a graph is symmetric.