On Weierstrass semigroups arising from finite graphs

In 2007, Baker and Norine proved an analogue of the Riemann-Roch Theorem for finite graphs. Motivated by the Weierstrass semigroup of a point on a nonsingular projective curve X, it is natural to consider analogues for a vertex P on a finite, connected graph which has no loops. Let Hr(P ) be the set of nonnegative integers α such that r(αP ) = r((α − 1)P ) + 1, wherer(D) denotes the dimension of a divisor D, and let Hf (P ) be the set of nonnegative integers α such that there exists an integer-valued function f on the vertices of a graph so that f has a pole only at P of order α. If P is a point on a curve X and r(αP ) is taken to be the dimension of the divisor αP , then these two sets are equal; indeed, Hr(P ) is well-studied Weierstass semigroup of P . However, in the case where P is a vertex of a finite graph G, these two sets may not be equal. In this paper, we explore the relationship between these sets.