Gluing of graphs and their Jacobians

Chilelli, Alessandro; Jun, Jaiung

doi:10.2140/involve.2023.16.389

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Gluing of graphs and their Jacobians

Alessandro Chilelli and Jaiung Jun

Vol. 16 (2023), No. 3, 389–407

DOI: 10.2140/involve.2023.16.389

Abstract

The Jacobian of a graph is a discrete analogue of the Jacobian of a Riemann surface. 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 (Exp. Math. 24:1 (2015), 1–7), stating it is affirmative in this case.

Keywords

Jacobian of a graph, sandpile group, critical group, chip-firing game, gluing graphs, cycle graph, Tutte polynomial, Tutte's rotor construction

Mathematical Subject Classification

Primary: 05C50, 05C76

Milestones

Received: 21 March 2021

Revised: 9 March 2022

Accepted: 13 June 2022

Published: 10 August 2023

Communicated by Kenneth S. Berenhaut

Authors

Alessandro Chilelli
Department of Mathematics State University of New York at Albany Albany, NY United States

Jaiung Jun
Department of Mathematics State University of New York at New Paltz New Paltz, NY United States

Vol. 16, No. 3, 2023