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

Abstract

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 In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of Delta , and so no distance-regular graph with intersection array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 }.