Association Schemes with Multiple Q-polynomial Structures
Hiroshi Suzuki
DOI: 10.1023/A:1008612505738
Abstract
It is well known that an association scheme X = ( X,{ R i } 0 \leqslant i \leqslant d ) \mathcal{X} = (X,\{ R_i \} _{0 \leqslant i \leqslant d} ) with $$ " align="middle" border="0"> has at most two P-polynomial structures. The parametrical condition for an association scheme to have two P-polynomial structures is also known. In this paper, we give a similar result for Q-polynomial association schemes. In fact, if 
" align="middle" border="0"> has at most two P-polynomial structures. The parametrical condition for an association scheme to have two P-polynomial structures is also known. In this paper, we give a similar result for Q-polynomial association schemes. In fact, if PDF
" align="middle" border="0"> has at most two P-polynomial structures. The parametrical condition for an association scheme to have two P-polynomial structures is also known. In this paper, we give a similar result for Q-polynomial association schemes. In fact, if PDF
References
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2. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin-Cummings, California, 1984.
3. A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer Verlag, Berlin, Heidelberg, 1989.
4. P. J. Cameron, J. M. Goethals and J. J. Seidel, “The Krein condition, spherical designs, Norton algebras and permutation groups,” Indag. Math. 40 (1978), 196-206.
5. G. A. Dickie, “Q-polynomial structures for association schemes and distance-regular graphs,” Ph.D. Thesis, University of Wisconsin, 1995.
6. G. A. Dickie, “A note on Q-polynomial association schemes,” preprint.
7. H. Suzuki, “A note on association schemes with two P-polynomial structures of type III,” J. Combin. Th. (A) 74 (1996), 158-168.
8. H. Suzuki, “On distance-i-graphs of distance-regular graphs,” Kyushu J. of Math. 48 (1994), 379-408.
9. H. Suzuki, “Imprimitive Q-polynomial association schemes,” J. Alg. Combin. 7 (1998), 165-180.
10. P. Terwilliger, “A characterization of the P - and Q-polynomial association schemes,” J. Combin. Th. (A) 45 (1987), 8-26.
11. P. Terwilliger, “Balanced sets and Q-polynomial association schemes,” Graphs and Combin. 4 (1988), 87-94.
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