Bipartite distance-regular graphs and taut pairs of pseudo primitive idempotents
Algebraic Combinatorics, Volume 2 (2019) no. 4, p. 499-520

Let Γ denote a bipartite distance-regular graph with diameter D4, valency k3, and intersection numbers c i ,b i (0iD). By a pseudo cosine sequence of Γ we mean a sequence of complex scalars σ 0 ,σ 1 ,...,σ D such that σ 0 =1 and c i σ i-1 +b i σ i+1 =kσ 1 σ i for 1iD-1. By an associated pseudo primitive idempotent of Γ, we mean a nonzero scalar multiple of the matrix i=0 D σ i A i , where A 0 ,A 1 ,...,A D are the distance matrices of Γ. Given pseudo primitive idempotents E,F of Γ, we define the pair E,F to be taut whenever the entry-wise product EF is not a scalar multiple of a pseudo primitive idempotent, but is a linear combination of two pseudo primitive idempotents of Γ. In this paper, we determine all the taut pairs of pseudo primitive idempotents of Γ.

Received : 2018-05-08
Revised : 2018-11-01
Accepted : 2018-11-24
Published online : 2019-08-01
DOI : https://doi.org/10.5802/alco.51
Classification:  05E30
Keywords: distance-regular graph, pseudo primitive idempotent, taut pair
@article{ALCO_2019__2_4_499_0,
     author = {MacLean, Mark S. and Miklavi\v c, \v Stefko},
     title = {Bipartite distance-regular graphs and taut pairs of pseudo primitive idempotents},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {4},
     year = {2019},
     pages = {499-520},
     doi = {10.5802/alco.51},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_4_499_0}
}
Bipartite distance-regular graphs and taut pairs of pseudo primitive idempotents. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 499-520. doi : 10.5802/alco.51. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_499_0/

[1] Bannai, Eiichi; Ito, Tatsuro Algebraic Combinatorics I, Association Schemes, Benjamin/Cummings Publishing Company, Inc, Mathematics lecture note series, Volume 58 (1984) | Zbl 0555.05019

[2] Brouwer, Andries E.; Cohen, Arjeh M.; Neumaier, Arnold Distance-Regular Graphs, Springer-Verlag, A Series of Modern Surveys in Mathematics (1989) | Zbl 0747.05073

[3] Curtin, Brian 2-homogeneous bipartite distance-regular graphs, Discrete Mathematics, Volume 187 (1998) no. 1-3, pp. 39-70 | Article | MR 1630676 | Zbl 0958.05143

[4] Curtin, Brian Almost 2-homogeneous bipartite distance-regular graphs, European Journal of Combinatorics, Volume 21 (2000) no. 7, pp. 865-876 | Article | MR 1787900 | Zbl 1002.05069

[5] Jurišić, Aleksandar; Terwilliger, Paul Pseudo 1-homogeneous distance-regular graphs, Journal of Algebraic Combinatorics, Volume 28 (2008) no. 4, pp. 509-529 | Article | MR 2443284 | Zbl 1160.05060

[6] Lang, Michael S Pseudo primitive idempotents and almost 2-homogeneous bipartite distance-regular graphs, European Journal of Combinatorics, Volume 29 (2008) no. 1, pp. 35-44 | Article | MR 2368612 | Zbl 1133.05105

[7] Lang, Michael S Bipartite distance-regular graphs: the Q-polynomial property and pseudo primitive idempotents, Discrete Mathematics, Volume 331 (2014), pp. 27-35 | Article | MR 3225303 | Zbl 1297.05264

[8] Maclean, Mark S Taut distance-regular graphs of odd diameter, Journal of Algebraic Combinatorics, Volume 17 (2003) no. 2, pp. 125-147 | MR 1971742 | Zbl 1014.05072

[9] Maclean, Mark S The local eigenvalues of a bipartite distance-regular graph, European Journal of Combinatorics, Volume 45 (2015), pp. 115-123 | Article | MR 3286625 | Zbl 1304.05095

[10] Maclean, Mark S; Miklavič, Štefko; Penjić, Safet On the Terwilliger algebra of bipartite distance-regular graphs with Δ 2 =0 and c 2 =1, Linear Algebra and its Applications, Volume 496 (2016), pp. 307-330 | Article | Zbl 1331.05237

[11] Maclean, Mark S; Terwilliger, Paul Taut distance-regular graphs and the subconstituent algebra, Discrete mathematics, Volume 306 (2006) no. 15, pp. 1694-1721 | Article | MR 2251102 | Zbl 1100.05104

[12] Pascasio, Arlene A; Terwilliger, Paul The pseudo-cosine sequences of a distance-regular graph, Linear algebra and its applications, Volume 419 (2006) no. 2-3, pp. 532-555 | Article | MR 2277986 | Zbl 1110.05105

[13] Penjić, Safet On the Terwilliger algebra of bipartite distance-regular graphs with Δ 2 =0 and c 2 =2, Discrete Mathematics, Volume 340 (2017) no. 3, pp. 452-466 | Article | MR 3584832 | Zbl 1351.05066

[14] Terwilliger, Paul; Weng, Chih-Wen Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra, European Journal of Combinatorics, Volume 25 (2004) no. 2, pp. 287-298 | Article | MR 2070549 | Zbl 1035.05104