Paper Info
Title
The uniqueness of a distance-regular graph with intersection array $$\\{32,27,8,1;1,4,27,32\\}$${32,27,8,1ź1,4,27,32} and related results
Abstract
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 In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of $$\\Delta $$Δ, and so no distance-regular graph with intersection array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^.
Year
DOI
Venue
2017
10.1007/s10623-016-0223-6
Designs, Codes and Cryptography
Keywords
Field
DocType
Distance-regular graph,Strongly regular graph,Antipodal cover,Fundamental group
Discrete mathematics,Graph,Uniqueness,Combinatorics,Strongly regular graph,Fundamental group,Isomorphism,Distance-regular graph,Antipodal point,Mathematics
Journal
Volume
Issue
ISSN
84
1-2
0925-1022
Citations 
PageRank 
References 
0
0.34
4
Authors
1
Name
Order
Citations
PageRank
Leonard H. Soicher14810.38