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
