Abstract
Let (X, G) be an association scheme. We say that (X, G) is flat if it is homogeneous and if any two distinct points have at most one common g-neighbor for each g ∈ G. In this paper we prove that any nondiscrete equitable partition of (X, G) has at most one singleton if (X, G) is flat, and {X} is the unique equitable partition without any singleton if (X, G) is flat and |X| is a prime.
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Bannai, E., Ito, T.: Algebraic Combinatorics I:Association Schemes, Benjamin/ Cummings, Menlo Park, CA, (1984)
Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, (1989)
Godsil, C.D., Martin, W.: Quotients of association schemes. J. Combin. Theory Ser. A 69(2), 185–199 (1995)
Ponomarenko, I.: in private communication
Zieschang, P.-H.: An Algebraic Approach to Association Schemes, Lecture Notes in Mathematics 1628. Springer, (1996)
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This work was supported for two years by Korea Research Foundation Grant (KRF-2006-003-C00010) and Pusan National University Research Grant.
Received: January 31, 2007. Final version received: Novmeber 14, 2007.
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Hirasaka, M., Kang, H. & Kim, K. Equitable Partitions of Flat Association Schemes. Graphs and Combinatorics 24, 81–89 (2008). https://doi.org/10.1007/s00373-008-0777-9
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DOI: https://doi.org/10.1007/s00373-008-0777-9