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In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like "pegging moves" allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging number (respectively, the optimal pegging number) of a graph is the minimum number of pegs such that for every (respectively, some) distribution of that many pegs on the graph, any vertex can be reached by a sequence of pegging moves. We prove several basic properties of pegging and analyze the pegging number and optimal pegging number of several classes of graphs, including paths, cycles, products with complete graphs, hypercubes, and graphs of small diameter.

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Let Γ be a quiver on n vertices v 1 , v 2 , . . . , v n with g ij edges between v i and v j , and let α ∈ N n . Hua gave a formula for A Γ (α, q), the number of isomorphism classes of absolutely indecomposable representations of Γ over the finite field F q with dimension vector α. Kac showed that A Γ (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 A Γ (α, 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.