  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.
