On Laplacian Monopoles

We consider the action of the (combinatorial) Laplacian of a finite and simple graph on integer vectors. By a Laplacian monopole we mean an image vector negative at exactly one coordinate associated with a vertex. We consider a numerical semigroup H-f(P) given by all monopoles at a vertex of a graph. The well-known analogy between finite graphs and algebraic curves (Riemann surfaces) has motivated much work. More specifically for us, the motivation arises out of the classical Weierstrass semigroup of a rational point on a curve whose properties are tied to the Riemann-Roch Theorem, as well as out of the graph theoretic Riemann-Roch Theorem demonstrated by Baker and Norine. We determine H-f(P) for some families of graphs and demonstrate a connection between H-f(P) and the vertex (also edge) connectivity of a graph. We also study H-r (P), another numerical semigroup which arises out of the result of Baker and Norine, and explore its connection to H-f(P) on graphs. We show that H-r(P) subset of H-f(P) in a number of special cases. In contrast to the situation in the classical setting, we demonstrate that H-f(P) \ H-r(P) can be arbitrarily large and identify a potential obstruction to the inclusion of H-r(P) in H-f(P) in general, though we still conjecture this inclusion. We conclude with a few open questions.