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.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.5144
 Bibcode:
 2014arXiv1410.5144G
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Geometry
 EPrint:
 18 pages, 8 TikZ figures. v2: Main results strengthened. Final version to appear in the Annals of Combinatorics