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The uniqueness of a distance-regular graph with intersection array \(\{32,27,8,1;1,4,27,32\}\) and related results. (English) Zbl 1367.05060

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 In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of \(\Delta \), and so no distance-regular graph with intersection array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

MSC:

05C12 Distance in graphs
05E45 Combinatorial aspects of simplicial complexes
55-04 Software, source code, etc. for problems pertaining to algebraic topology
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