Exponents of Jacobians of Graphs and Regular Matroids

TL;DR

This paper studies the Jacobian groups of graphs and regular matroids, proving finiteness results for those with specific exponents and characterizing all such cases for exponent 2.

Contribution

It extends the concept of Jacobians from graphs to regular matroids and characterizes all connected regular matroids with Jacobian exponent 2.

Findings

Finitely many biconnected graphs have Jacobian exponent 2 or 3.

Characterization of all connected regular matroids with Jacobian exponent 2.

Abstract

Let $G$ be a finite undirected multigraph with no self-loops. The Jacobian $\operatorname{Jac}(G)$ is a finite abelian group associated with $G$ whose cardinality is equal to the number of spanning trees of $G$. There are only a finite number of biconnected graphs $G$ such that the exponent of $\operatorname{Jac}(G)$ equals $2$ or $3$. The definition of a Jacobian can also be extended to regular matroids as a generalization of graphs. We prove that there are finitely many connected regular matroids $M$ such that $\operatorname{Jac}(M)$ has exponent $2$ and characterize all such matroids.