## 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:

 5e+30 Association schemes, strongly regular graphs

### Keywords:

association scheme; Terwilliger algebra

### Citations:

Zbl 0785.05089; Zbl 1001.05124
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.