Counting quiver representations over finite fields via graph enumeration

Let Γ be a quiver on n vertices v1,v2,…,vn with gij edges between vi and vj, and let α∈Nn. Hua gave a formula for AΓ(α,q), the number of isomorphism classes of absolutely indecomposable representations of Γ over the finite field Fq with dimension vector α. Kac showed that AΓ(α,q) is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each integer s⩾0, the sth derivative of AΓ(α,q) with respect to q, when evaluated at q=1, is a polynomial in the variables gij, 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.