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Gluing of graphs and their Jacobians

Alessandro Chilelli and Jaiung Jun

Vol. 16 (2023), No. 3, 389–407
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.

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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