# zbMATH — the first resource for mathematics

Bipartite distance-regular graphs: the $$Q$$-polynomial property and pseudo primitive idempotents. (English) Zbl 1297.05264
Summary: Let $$\Gamma$$ denote a bipartite distance-regular graph with diameter at least 4 and valency at least 3. Fix a vertex of $$\Gamma$$ and let $$T$$ denote the corresponding Terwilliger algebra. Suppose that $$\Gamma$$ is $$Q$$-polynomial and there are two non-isomorphic irreducible $$T$$-modules with endpoint 2. We show that, unless the intersection numbers of $$\Gamma$$ fit one exceptional case (which is not known to correspond to an actual graph), the entry-wise product of pseudo primitive idempotents associated with these modules is a linear combination of two pseudo primitive idempotents. This result relates to a conjecture of MacLean and Terwilliger.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
Full Text:
##### References:
 [1] Bannai, Eiichi; Ito, Tatsuro, Algebraic combinatorics. I. association schemes, (1984), The Benjamin/Cummings Publishing Co. Inc. Menlo Park, CA · Zbl 0555.05019 [2] Biggs, Norman, Algebraic graph theory, Cambridge Mathematical Library, (1993), Cambridge University Press Cambridge · Zbl 0797.05032 [3] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., (Distance-Regular Graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18, (1989), Springer-Verlag Berlin) · Zbl 0747.05073 [4] Caughman, John S., The Terwilliger algebras of bipartite $$P$$- and $$Q$$-polynomial schemes, Discrete Math., 196, 1-3, 65-95, (1999) · Zbl 0924.05067 [5] Curtin, Brian, Almost 2-homogeneous bipartite distance-regular graphs, European J. Combin., 21, 7, 865-876, (2000) · Zbl 1002.05069 [6] Lang, Michael S., Tails of bipartite distance-regular graphs, European J. Combin., 23, 8, 1015-1023, (2002) · Zbl 1012.05159 [7] Lang, Michael S., A new inequality for bipartite distance-regular graphs, J. Combin. Theory Ser. B, 90, 1, 55-91, (2004) · Zbl 1051.05082 [8] MacLean, Mark S.; Terwilliger, Paul, Taut distance-regular graphs and the subconstituent algebra, Discrete Math., 306, 15, 1694-1721, (2006) · Zbl 1100.05104 [9] Nomura, K., Homogeneous graphs and regular near polygons, J. Combin. Theory Ser. B, 60, 1, 63-71, (1994) · Zbl 0793.05130 [10] Nomura, K., Spin models on bipartite distance-regular graphs, J. Combin. Theory Ser. B, 64, 2, 300-313, (1995) · Zbl 0827.05060 [11] Pascasio, Arlene A.; Terwilliger, Paul, The pseudo-cosine sequences of a distance-regular graph, Linear Algebra Appl., 419, 2-3, 532-555, (2006) · Zbl 1110.05105 [12] Terwilliger, Paul, The subconstituent algebra of an association scheme. I, J. Algebraic Combin., 1, 4, 363-388, (1992) · Zbl 0785.05089 [13] Terwilliger, Paul, The subconstituent algebra of an association scheme. II, J. Algebraic Combin., 2, 1, 73-103, (1993) · Zbl 0785.05090 [14] Terwilliger, Paul; Weng, Chih-Wen, Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra, European J. Combin., 25, 2, 287-298, (2004) · Zbl 1035.05104 [15] Janoš Vidali, There is no distance-regular graph with intersection array 55,54,50,35,10;1,5,20,45,55. Preprint.
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.