The graphs of Hoffman-Singleton, Higman-Sims, and McLaughlin, and the Hermitian curve of degree 6 in characteristic 5

TL;DR

This paper constructs specific well-known graphs from geometric relations on Hermitian curves in characteristic 5 and interprets these constructions through automorphism group subgroup structures.

Contribution

It provides a novel geometric construction of the Hoffman-Singleton, Higman-Sims, and McLaughlin graphs using Hermitian curves and relates these to automorphism groups.

Findings

Constructed graphs from Hermitian curve relations

Connected graph structures to automorphism subgroup structures

Enhanced understanding of geometric origins of these graphs

Abstract

We construct the graphs of Hoffman-Singleton, Higman-Sims, and McLaughlin from certain relations on the set of non-singular conics totally tangent to the Hermitian curve of degree 6 in characteristic 5. We then interpret this geometric construction in terms of the subgroup structure of the automorphism group of this Hermitian curve.