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Annals of Combinatorics

, Volume 22, Issue 4, pp 781–801 | Cite as

Realization of Groups with Pairing as Jacobians of Finite Graphs

  • Louis Gaudet
  • David Jensen
  • Dhruv Ranganathan
  • Nicholas Wawrykow
  • Theodore Weisman
Article
  • 8 Downloads

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.

Keywords

Graph Jacobians Groups with pairing 

Mathematics Subject Classification

05C25 14T05 

Notes

Acknowledgements

This project was completed as part of the 2014 Summer Undergraduate Mathematics Research at Yale (SUMRY) program, where the second and third authors were supported as mentors and the first, fourth, and fifth authors were supported as participants. It is a pleasure to thank all involved in the program for creating a vibrant research community. We benefited from conversations with Dan Corey, Andrew Deveau, Jenna Kainic, Nathan Kaplan, Susie Kimport, Dan Mitropolsky, and Anup Rao. We thank Sam Payne for suggesting the problem. We are also especially grateful to Paul Pollack, whose ideas significantly strengthened the results of this paper. Finally, we thank the referees for their careful reading and insightful comments. The authors were supported by NSF grant CAREER DMS-1149054 (PI: Sam Payne).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Louis Gaudet
    • 2
  • David Jensen
    • 1
  • Dhruv Ranganathan
    • 3
  • Nicholas Wawrykow
    • 2
  • Theodore Weisman
    • 2
  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA
  3. 3.Department of MathematicsMassachusetts Institute of Technology CambridgeUSA

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