Realization of groups with pairing as Jacobians of finite graphs

TL;DR

This paper investigates which finite abelian groups with pairing can be realized as Jacobians of finite graphs, providing explicit constructions and conditional results based on number theory conjectures.

Contribution

It offers explicit graph constructions for realizing large classes of groups with pairing as Jacobians, advancing understanding of the inverse problem in graph theory.

Findings

Constructed graphs realize a large fraction of odd groups with pairing.

Conditional on the Riemann hypothesis, all odd order groups with pairing are realizable.

Identified limitations for simple graphs, showing some groups cannot be realized.

Abstract

We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs.