On Laplacian Monopoles

TL;DR

This paper explores the structure of Laplacian monopoles on graphs, connecting graph theory with algebraic geometry concepts, and investigates the properties of associated numerical semigroups and their relations.

Contribution

It determines the Laplacian monopole semigroup for certain graph families and analyzes the relationship between two types of semigroups, revealing new insights into graph connectivity and algebraic analogies.

Findings

Determined $H_f(P)$ for specific graph families.

Established connections between $H_f(P)$, $H_r(P)$, and graph connectivity.

Showed that $H_f(P)\setminus H_r(P)$ can be arbitrarily large.

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.