Critical group structure from the parameters of a strongly regular graph

TL;DR

This paper establishes arithmetic conditions based on strongly regular graph parameters that determine the structure of the graph's critical group, providing new tools for analyzing graph invariants and applying to specific examples.

Contribution

It introduces simple arithmetic conditions linking strongly regular graph parameters to the structure of their critical groups, enabling explicit computations and non-existence proofs.

Findings

Determined the Sylow p-subgroup structure of critical groups for certain strongly regular graphs.

Applied the theory to compute the critical group of Conway's 99-graph.

Provided an elementary proof of the non-existence of srg(28,9,0,4).

Abstract

We give simple arithmetic conditions that force the Sylow $p$-subgroup of the critical group of a strongly regular graph to take a specific form. These conditions depend only on the parameters $(v, k, \lambda, \mu)$ of the strongly regular graph under consideration. We give many examples, including how the theory can be used to compute the critical group of Conway's $99$-graph and to give an elementary argument that no $srg(28,9,0,4)$ exists.