Critical Groups of Graphs with Dihedral Actions

TL;DR

This paper studies the structure of the critical group of finite connected graphs with dihedral symmetry, showing how it decomposes based on the graph's quotient structures, paralleling a known algebraic curve decomposition.

Contribution

It introduces a decomposition theorem for the critical group of graphs with dihedral automorphisms, extending concepts from algebraic geometry to graph theory.

Findings

Critical group decomposes in terms of quotients by subgroups of dihedral automorphisms.

Applicable when all orbits have size n or 2n.

Analogy with Jacobian decomposition of algebraic curves.

Abstract

In this paper we consider the critical group of finite connected graphs which admit harmonic actions by the dihedral group $D_n$. In particular, we show that if the orbits of the $D_n$-action all have either $n$ or $2n$ points then the critical group of such a graph can be decomposed in terms of the critical groups of the quotients of the graph by certain subgroups of the automorphism group. This is analogous to a theorem of Kani and Rosen which decomposes the Jacobians of algebraic curves with a $D_n$-action.