The uniqueness of a distance-regular graph with intersection array $${32,27,8,1;1,4,27,32}$$ and related results

It is known that, up to isomorphism, there is a unique distance-regular graph (Delta ) with intersection array ({32,27;1,12}) [equivalently, (Delta ) is the unique strongly regular graph with parameters (105, 32, 4, 12)]. Here we investigate the distance-regular antipodal covers of (Delta ). We show that, up to isomorphism, there is just one distance-regular antipodal triple cover of (Delta ) (a graph (hat{Delta }) discovered by the author over 20 years ago), proving that there is a unique distance-regular graph with intersection array ({32,27,8,1;1,4,27,32}). In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of (Delta ), and so no distance-regular graph with intersection array ({32,27,6,1;1,6,27,32}). We also show there is no distance-regular antipodal 4-cover of (Delta ), and so no distance-regular graph with intersection array ({32,27,9,1;1,3,27,32}), and that there is no distance-regular antipodal 6-cover of (Delta ) that is a double cover of (hat{Delta }).