Let G be a finite undirected multigraph with no self-loops. The Jacobian 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 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 Jac (M) has exponent 2 and characterize all such matroids.
Lheem, Hahn; Li, Deyuan; Quines, Carl Joshua; and Zhang, Jessica
"Exponents of Jacobians of Graphs and Regular Matroids,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 21
, Article 6.
Available at: https://scholar.rose-hulman.edu/rhumj/vol21/iss2/6