Realization of Groups with Pairing as Jacobians of Finite Graphs
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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.
KeywordsGraph Jacobians Groups with pairing
Mathematics Subject Classification05C25 14T05
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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