A Torelli Theorem for Graph Isomorphisms

TL;DR

This paper characterizes when cyclic bijections induced by graph Jacobians can be extended to full graph isomorphisms, refining algebraic geometry tools for analyzing the Torelli map's fibers.

Contribution

It provides a criterion based on an easily computed divisor to determine when cyclic bijections correspond to actual graph isomorphisms.

Findings

Provides a necessary and sufficient condition for cyclic bijections to be graph isomorphisms.

Refines algebraic geometry methods for studying the Torelli map.

Enhances understanding of the relationship between graph Jacobians and graph isomorphisms.

Abstract

It is known that isomorphisms of graph Jacobians induce cyclic bijections on the associated graphs. We characterize when such cyclic bijections can be strengthened to graph isomorphisms, in terms of an easily computed divisor. The result refines tools used in algebraic geometry to examine the fibers of the compactified Torelli map.