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It is known that, up to isomorphism, there is a unique distance-regular graph $$\\Delta $$Δ with intersection array $$\\{32,27;1,12\\}$${32,27ź1,12} [equivalently, $$\\Delta $$Δ is the unique strongly regular graph with parameters (105, 32, 4, 12)]. Here we investigate the distance-regular antipodal covers of $$\\Delta $$Δ. We show that, up to isomorphism, there is just one distance-regular antipodal triple cover of $$\\Delta $$Δ (a graph $$\\hat{\\Delta }$$Δ^ discovered by the author over 20 years ago), proving that there is a unique distance-regular graph with intersection array $$\\{32,27,8,1;1,4,27,32\\}$${32,27,8,1ź1,4,27,32}. In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of $$\\Delta $$Δ, and so no distance-regular graph with intersection array $$\\{32,27,6,1;1,6,27,32\\}$${32,27,6,1ź1,6,27,32}. We also show there is no distance-regular antipodal 4-cover of $$\\Delta $$Δ, and so no distance-regular graph with intersection array $$\\{32,27,9,1;1,3,27,32\\}$${32,27,9,1ź1,3,27,32}, and that there is no distance-regular antipodal 6-cover of $$\\Delta $$Δ that is a double cover of $$\\hat{\\Delta }$$Δ^.

designs, codes and cryptography

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