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.
@article{10_37236_8954,
author = {Robert F. Bailey and Daniel R. Hawtin},
title = {On the 486-vertex distance-regular graphs of {Koolen-Riebeek} and {Soicher}},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {3},
doi = {10.37236/8954},
zbl = {1444.05153},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8954/}
}
TY - JOUR
AU - Robert F. Bailey
AU - Daniel R. Hawtin
TI - On the 486-vertex distance-regular graphs of Koolen-Riebeek and Soicher
JO - The electronic journal of combinatorics
PY - 2020
VL - 27
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/8954/
DO - 10.37236/8954
ID - 10_37236_8954
ER -
%0 Journal Article
%A Robert F. Bailey
%A Daniel R. Hawtin
%T On the 486-vertex distance-regular graphs of Koolen-Riebeek and Soicher
%J The electronic journal of combinatorics
%D 2020
%V 27
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/8954/
%R 10.37236/8954
%F 10_37236_8954
Robert F. Bailey; Daniel R. Hawtin. On the 486-vertex distance-regular graphs of Koolen-Riebeek and Soicher. The electronic journal of combinatorics, Tome 27 (2020) no. 3. doi: 10.37236/8954
