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Taut distance-regular graphs and the subconstituent algebra. (English) Zbl 1100.05104
This paper deals with bipartite distance-regular graphs \(\Gamma \) with diameter \(D\geq 4\), valency \(k\geq 3\) and intersection numbers \(a_{i},b_{i},c_{i}\). The authors obtain some results on the subconstituent algebra of \(\Gamma \) introduced by P. Terwilliger [J. Algebr. Comb. 1, 363–388 (1992; Zbl 0785.05089)], and some related results concerning the taut condition proposed by M. S. MacLean [Discrete Math. 225, 193–216 (2000; Zbl 1001.05124)]. Upper and lower bounds for the local eigenvalues are obtained in terms of the intersection numbers of \(\Gamma \) and the eigenvalues of the adjacency matrix of \(\Gamma \). A detailed description of the thin irreducible \(T\)-modules that have endpoint 2 and dimension \(D-3\) is given, where \(T=T(x)\) is the Terwilliger algebra of \(\Gamma \) with respect to vertex \(x\). Three characterizations of the taut condition are obtained, each of which involving the local eigenvalues or the above \(T\)-modules.

MSC:
05E30 Association schemes, strongly regular graphs
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[1] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings London · Zbl 0555.05019
[2] Biggs, N., Algebraic graph theory, (1994), Cambridge University Press London · Zbl 0797.05032
[3] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer Berlin · Zbl 0747.05073
[4] Caughman, J.S., The Terwilliger algebras of bipartite P- and Q-polynomial schemes, Discrete math., 196, 65-95, (1999) · Zbl 0924.05067
[5] Collins, B., The Terwilliger algebra of an almost-bipartite distance-regular graph and its antipodal 2-cover, Discrete math., 216, 35-69, (2000) · Zbl 0955.05113
[6] Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete math., 187, 39-70, (1998) · Zbl 0958.05143
[7] Curtin, B., Bipartite distance-regular graphs I, Graphs combin., 15, 143-158, (1999) · Zbl 0927.05083
[8] Curtin, B., Bipartite distance-regular graphs II, Graphs combin., 15, 377-391, (1999) · Zbl 0939.05088
[9] Curtin, B.; Nomura, K., Distance-regular graphs related to the quantum enveloping algebra of \(\mathit{sl}(2)\), J. algebraic combin., 12, 25-36, (2000) · Zbl 0967.05067
[10] Curtis, C.; Reiner, I., Representation theory of finite groups and associative algebras, (1962), Interscience New York · Zbl 0131.25601
[11] Dickie, G., Twice \(Q\)-polynomial distance-regular graphs are thin, European J. combin., 16, 555-560, (1995) · Zbl 0852.05085
[12] Egge, E., A generalization of the Terwilliger algebra, J. algebra, 233, 213-252, (2000) · Zbl 0960.05108
[13] Go, J.T., The Terwilliger algebra of the hypercube, European J. combin., 23, 399-429, (2002) · Zbl 0997.05097
[14] Go, J.T.; Terwilliger, P., Tight distance-regular graphs and the subconstituent algebra, European J. combin., 23, 793-816, (2002) · Zbl 1014.05070
[15] Godsil, C.D., Algebraic combinatorics, (1993), Chapman & Hall Inc. New York · Zbl 0814.05075
[16] Hobart, S.A.; Ito, T., The structure of nonthin irreducible T-modules: ladder bases and classical parameters, J. algebraic combin., 7, 53-75, (1998) · Zbl 0911.05059
[17] Jurišić, A.; Koolen, J., A local approach to 1-homogeneous graphs, Des. codes cryptogr., 21, 127-147, (2000) · Zbl 0964.05073
[18] Jurišić, A.; Koolen, J., Nonexistence of some antipodal distance-regular graphs of diameter four, European J. combin., 21, 1039-1046, (2000) · Zbl 0958.05139
[19] Jurišić, A.; Koolen, J., Krein parameters and antipodal distance-regular graphs with diameter 3 and 4, Discrete math., 244, 181-202, (2002) · Zbl 1024.05086
[20] Jurišić, A.; Koolen, J., 1-homogeneous graphs with cocktail party \(\mu\)-graphs, J. algebraic combin., 18, 79-98, (2003) · Zbl 1038.05059
[21] Jurišić, A.; Koolen, J.; Terwilliger, P., Tight distance-regular graphs, J. algebraic combin., 12, 163-197, (2000) · Zbl 0959.05121
[22] MacLean, M., An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete math., 225, 193-216, (2000) · Zbl 1001.05124
[23] MacLean, M., Taut distance-regular graphs of odd diameter, J. algebraic combin., 17, 125-147, (2003) · Zbl 1014.05072
[24] MacLean, M., Taut distance-regular graphs of even diameter, J. combin. theory ser. B, 91, 127-142, (2004) · Zbl 1047.05044
[25] Nomura, K., Homogeneous graphs and regular near polygons, J. combin. theory ser. B, 60, 63-71, (1994) · Zbl 0793.05130
[26] Nomura, K., Spin models on bipartite distance-regular graphs, J. combin. theory ser. B, 64, 300-313, (1995) · Zbl 0827.05060
[27] K. Nomura, Spin models and almost bipartite 2-homogeneous graphs, Advanced Studies in Pure Mathematics, vol. 24, Mathematical Society Japan,Tokyo, 1996, pp. 285-308. · Zbl 0858.05101
[28] Pascasio, A.A., Tight graphs and their primitive idempotents, J. algebraic combin., 10, 47-59, (1999) · Zbl 0927.05085
[29] Pascasio, A.A., Tight distance-regular graphs and Q-polynomial property, Graphs combin., 17, 149-169, (2001) · Zbl 0993.05147
[30] Pascasio, A.A., An inequality on the cosines of a tight distance-regular graph, Linear algebra appl., 325, 147-159, (2001) · Zbl 0979.05112
[31] Pascasio, A.A., An inequality in character algebras, Discrete math., 264, 201-210, (2003) · Zbl 1014.05076
[32] A.A. Pascasio, P. Terwilliger, The pseudocosine sequences of a distance-regular graph, Linear Algebra Appl., submitted. · Zbl 1110.05105
[33] Tanabe, K., The irreducible modules of the Terwilliger algebras of Doob schemes, J. algebraic combin., 6, 173-195, (1997) · Zbl 0868.05056
[34] Terwilliger, P., The subconstituent algebra of an association scheme I, J. algebraic combin., 1, 363-388, (1992) · Zbl 0785.05089
[35] Terwilliger, P., The subconstituent algebra of an association scheme II, J. algebraic combin., 2, 73-103, (1993) · Zbl 0785.05090
[36] Terwilliger, P., The subconstituent algebra of an association scheme III, J. algebraic combin., 2, 177-210, (1993) · Zbl 0785.05091
[37] Terwilliger, P., The subconstituent algebra of a distance-regular graph; thin modules with endpoint one, Linear algebra appl., 356, 157-187, (2002) · Zbl 1011.05066
[38] Terwilliger, P.; Weng, C.-W., Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra, European J. combin., 25, 287-298, (2004) · Zbl 1035.05104
[39] Tomiyama, M., On the primitive idempotents of distance-regular graphs, Discrete math., 240, 281-294, (2001) · Zbl 0993.05148
[40] Tomiyama, M.; Yamazaki, N., The subconstituent algebra of a strongly regular graph, Kyushu J. math., 48, 323-334, (1998) · Zbl 0842.05098
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