Bipartite distance-regular graphs and taut pairs of pseudo primitive idempotents

MacLean, Mark S.; Miklavič, Štefko

doi:10.5802/alco.51

MacLean, Mark S.¹; Miklavič, Štefko ²

¹ Mathematics Department Seattle University 901 Twelfth Avenue Seattle WA 98122-1090, USA
² University of Primorska Andrej Marušič Institute Muzejski trg 2 6000 Koper, Slovenia

Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 499-520.

Abstract

Let $Γ$ denote a bipartite distance-regular graph with diameter $D \geq 4$ , valency $k \geq 3$ , and intersection numbers $c_{i}, b_{i} (0 \leq i \leq D)$ . 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 $c_{i} σ_{i - 1} + b_{i} σ_{i + 1} = k σ_{1} σ_{i}$ for $1 \leq i \leq D - 1$ . By an associated pseudo primitive idempotent of $Γ$ , we mean a nonzero scalar multiple of the matrix $\sum_{i = 0}^{D} σ_{i} A_{i}$ , 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 $E \circ F$ 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

MR Zbl

DOI: 10.5802/alco.51

Classification: 05E30
Keywords: distance-regular graph, pseudo primitive idempotent, taut pair

Author's affiliations:

MacLean, Mark S. ¹; Miklavič, Štefko ²

¹ Mathematics Department Seattle University 901 Twelfth Avenue Seattle WA 98122-1090, USA
² University of Primorska Andrej Marušič Institute Muzejski trg 2 6000 Koper, Slovenia

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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MacLean, Mark S.; Miklavič, Štefko. 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/articles/10.5802/alco.51/

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