#### Abstract

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.

#### Author Bio

Hahn Lheem is an undergraduate student at Harvard University. Hahn attended the Program in Mathematics for Young Scientists at Boston University over the summers of 2017, 2018, and 2019. The research was conducted at the Program in Mathematics for Young Scientists in 2019.

Deyuan Li is a student at Yale University. Deyuan attended the Program in Mathematics for Young Scientists at Boston University over the summers of 2018 and 2019. The research was conducted at the Program in Mathematics for Young Scientists in 2019.

Carl Joshua (CJ) Quines is an undergraduate student at the Massachusetts Institute of Technology. CJ attended the Program in Mathematics for Young Scientists at Boston University over the summers of 2018 and 2019. The research was conducted at the Program in Mathematics for Young Scientists in 2019.

Jessica Zhang is a student at Proof School. Jessica attended the Program in Mathematics for Young Scientists at Boston University over the summers of 2018 and 2019. The research was conducted at the Program in Mathematics for Young Scientists in 2019.

#### Recommended Citation

Lheem, Hahn; Li, Deyuan; Quines, Carl Joshua; and Zhang, Jessica
(2020)
"Exponents of Jacobians of Graphs and Regular Matroids,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 21
:
Iss.
2
, Article 6.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol21/iss2/6

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