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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 formulafor $A_{\Gamma}(\alpha, q)$, the number of isomorphism classes of absolutelyindecomposable representations of $\Gamma$ over the finite field $\F_q$ withdimension vector $\alpha$. Kac showed that $A_{\Gamma}(\bm{\alpha}, q)$ is apolynomial in q with integer coefficients. Using Hua's formula, we show thatfor 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. Ourformulas for these coefficients depend on the enumeration of certain familiesof connected graphs.

Counting Quiver Representations over Finite Fields Via Graph Enumeration