A Note on the Critical Groups of Strongly Regular Graphs and Their Generalizations

TL;DR

This paper investigates the maximum order of elements in the critical groups of strongly regular graphs, extends results to graphs with two non-zero Laplacian eigenvalues, and explores related algebraic structures.

Contribution

It determines the spectral bound for the critical group elements in strongly regular graphs and generalizes the result to specific classes of graphs, including signed graphs.

Findings

Maximum order of elements in critical groups matches spectral bounds

Extension of bounds to graphs with two non-zero Laplacian eigenvalues

Insights into the monodromy pairing and structure of critical groups

Abstract

We determine the maximum order of an element in the critical group of a strongly regular graph, and show that it achieves the spectral bound due to Lorenzini. We extend the result to all graphs with exactly two non-zero Laplacian eigenvalues, and study the signed graph version of the problem. We also study the monodromy pairing on the critical groups, and suggest an approach to study the structure of these groups using the pairing.