On the 486-vertex distance-regular graphs of Koolen--Riebeek and Soicher

TL;DR

This paper analyzes three specific 486-vertex distance-regular graphs, revealing their symmetries and connections to the ternary Golay code, and classifies their automorphism groups.

Contribution

It identifies a common automorphism group action on three imprimitive distance-regular graphs and explains their connection via the ternary Golay code.

Findings

All three graphs are preserved by the same rank-9 group action.

The graphs are related to the affine geometry AG(5,3).

The connection to the ternary Golay code is established.

Abstract

This paper considers three imprimitive distance-regular graphs with 486 vertices and diameter 4: the Koolen--Riebeek graph (which is bipartite), the Soicher graph (which is antipodal), and the incidence graph of a symmetric transversal design obtained from the affine geometry $\mathrm{AG}(5,3)$ (which is both). It is shown that each of these is preserved by the same rank-9 action of the group $3^5:(2\times M_{10})$, and the connection is explained using the ternary Golay code.