On Laplacian Monopoles

Abstract

We consider the action of the (combinatorial) Laplacian of a finite and simple graph on integer vectors. By a \emph{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 $H_r(P)\subseteq H_f(P)$ in a number of special cases. In contrast to the situation in the classical setting, we demonstrate that $H_f(P)\setminus H_r(P)$ 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.