Gluing of graphs and their Jacobians

TL;DR

This paper investigates how graph Jacobians change under gluing operations, links their computation to cycle matrices, and demonstrates that certain planar graphs have isomorphic Jacobians, answering a specific open question.

Contribution

It introduces methods to analyze Jacobians under graph gluing, connects Jacobian computation to cycle matrices, and proves a case where Tutte's rotor construction yields isomorphic Jacobians.

Findings

Jacobians of glued graphs can be explicitly described.

Cycle matrices are instrumental in computing graph Jacobians.

Tutte's rotor construction produces planar graphs with isomorphic Jacobians.

Abstract

The Jacobian of a graph is a discrete analogue of the Jacobian of a Riemann surface. In this paper, we explore how Jacobians of graphs change when we glue two graphs along a common subgraph focusing on the case of cycle graphs. Then, we link the computation of Jacobians of graphs with cycle matrices. Finally, we prove that Tutte's rotor construction with his original example produces two graphs with isomorphic Jacobians when all involved graphs are planar. This answers the question posed by Clancy, Leake, and Payne, stating it is affirmative in this case.