Cyclic Critical Groups of Graphs

TL;DR

This paper introduces a construction method for graphs with cyclic critical groups, provides formulas for spanning trees in some cases, and discusses open questions in the area.

Contribution

It presents a new construction for graphs with cyclic critical groups and derives formulas for their spanning trees, advancing understanding of graph invariants.

Findings

Identifies families of graphs with cyclic critical groups

Provides formulas for the number of spanning trees in some cases

Poses open questions for future research

Abstract

In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the group exactly. We also pose several open questions related to this work.