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 (Delta) over cap 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 In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of Delta, and so no distance-regular graph with intersection array {32, 27, 6,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 (Delta) over cap.