Critical group structure from the parameters of a strongly regular graph

Joshua E. Ducey; David L. Duncan; Wesley J. Engelbrecht; Jawahar V. Madan; Eric Piato; Christina S. Shatford; Angela Vichitbandha

arXiv · 1910.07686

Critical group structure from the parameters of a strongly regular graph

Joshua E. Ducey·David L. Duncan·Wesley J. Engelbrecht·Jawahar V. Madan·Eric Piato·Christina S. Shatford·Angela Vichitbandha

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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 $(v, k, λ, μ)$ 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.

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Joshua E. Ducey, David L. Duncan, Wesley J. Engelbrecht, Jawahar V. Madan, Eric Piato, Christina S. Shatford, Angela Vichitbandha. 2019-10-17. Critical group structure from the parameters of a strongly regular graph. https://arxiv.org/abs/1910.07686

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