Counting Quiver Representations over Finite Fields Via Graph Enumeration
  Let $\Gamma$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\alpha \in \N^n$. Hua gave a formula for $A_{\Gamma}(\alpha, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\F_q$ with dimension vector $\alpha$. Kac showed that $A_{\Gamma}(\bm{\alpha}, q)$ is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each non-negative integer s, the s-th derivative of $A_{\Gamma}(\alpha,q)$ with respect to q, when evaluated at q = 1, is a polynomial in the variables $g_{ij}$, and we compute the highest degree terms in this polynomial. Our formulas for these coefficients depend on the enumeration of certain families of connected graphs. 
Geir T. Helleloid and Fernando Rodriguez Villegas