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.
Highlights
A graph with diameter d is distance-regular if, for all i with 0 i d and any vertices u, w at distance i, the number of neighbours of w at distances i − 1, i and i + 1 from u depends only on i, and not on the choices of u and w
Ci + ai + bi = k for 1 i d − 1 and cd + ad = k, so the parameters ai are determined by the others
To construct the Koolen–Riebeek graph ∆, we use the fact that Aut(Γ) ∼= 35 : (2 × M11) is a primitive permutation group of rank 3
Summary
A (finite, simple, connected) graph with diameter d is distance-regular if, for all i with 0 i d and any vertices u, w at distance i, the number of neighbours of w at distances i − 1, i and i + 1 from u depends only on i, and not on the choices of u and w. The ternary Golay code G is a 6-dimensional linear code in F131, which is the unique such perfect code with minimum distance 5 (see [15]); its coset graph is the Berlekamp–van Lint–Seidel graph, obtained in [3], which is a strongly regular graph Γ with parameters (243, 22, 1, 2). The Koolen–Riebeek graph is a bipartite distance-regular graph ∆ with 486 vertices, diameter 4 and intersection array
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