Critical groups of graphs with reflective symmetry

TL;DR

This paper generalizes a known factorization of the spanning tree number for graphs with reflective symmetry, revealing new structural insights into their critical groups through exact sequences and bicycle space connections.

Contribution

It introduces a new exact sequence relating the critical group of symmetric graphs to their substructures, extending previous results by Ciucu-Yan-Zhang.

Findings

Established an exact sequence for the critical group of symmetric graphs.

Connected the critical group structure to bicycle spaces.

Extended factorization results to a broader class of graphs.

Abstract

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. For a graph G with a certain reflective symmetry, we generalize a result of Ciucu-Yan-Zhang factorizing the spanning tree number of G by interpreting this as a result about the critical group of G. Our result takes the form of an exact sequence, and explicit connections to bicycle spaces are made.