Realization of groups with pairing as jacobians of finite graphs

Louis Gaudet; David Jensen; Dhruv Ranganathan; Nicholas Wawrykow; Theodore Weisman

doi:10.1007/s00026-018-0406-0

Realization of groups with pairing as jacobians of finite graphs

Louis Gaudet
, David Jensen
, Dhruv Ranganathan
, Nicholas Wawrykow
, Theodore Weisman

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	781-801
Number of pages	21
Journal	Annals of Combinatorics
Volume	22
Issue number	4
DOIs	https://doi.org/10.1007/s00026-018-0406-0
State	Published - Jan 1 2018

Bibliographical note

Publisher Copyright:
© 2018 Springer Nature Switzerland AG.

Funding

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 Kim-port, 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). 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).

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	CAREER DMS-1149054

Keywords

Graph jacobians
Groups with pairing

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

Access to Document

10.1007/s00026-018-0406-0

Cite this

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title = "Realization of groups with pairing as jacobians of finite graphs",

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.",

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AU - Weisman, Theodore

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N2 - 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.

AB - 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.

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Realization of groups with pairing as jacobians of finite graphs

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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