On the Graphs of Hoffman-Singleton and Higman-Sims

Paul R. Hafner


We propose a new elementary definition of the Higman-Sims graph
in which the 100 vertices are parametrised with $Z_4timesZ_5timesZ_5$
and adjacencies are described by linear and quadratic equations.
This definition extends Robertson's pentagon-pentagram
definition of the Hoffman-Singleton graph and is obtained
by studying maximum cocliques of the Hoffman-Singleton graph
in Robertson's parametrisation. The new description is used
to count the 704 Hoffman-Singleton subgraphs in the Higman-Sims
graph, and to describe the two orbits of the simple group HS
on them, including a description of the doubly transitive
action of HS within the Higman-Sims graph.
Numerous geometric connections are pointed out.

Hoffman-Singleton graph, Higman-Sims graph, Higman-Sims group, biaffine plane, S(3,6,22)

Math Review Classification
Primary 05C62, 05C25 ; Secondary 05B25, 51E10, 51E26

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