# ALGEBRAIC COMBINATORICS

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

Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D\ge 4$, valency $k\ge 3$, and intersection numbers ${c}_{i},{b}_{i}\phantom{\rule{0.277778em}{0ex}}\left(0\le i\le D\right)$. By a pseudo cosine sequence of $\Gamma$ we mean a sequence of complex scalars ${\sigma }_{0},{\sigma }_{1},...,{\sigma }_{D}$ such that ${\sigma }_{0}=1$ and ${c}_{i}{\sigma }_{i-1}+{b}_{i}{\sigma }_{i+1}=k{\sigma }_{1}{\sigma }_{i}$ for $1\le i\le D-1$. By an associated pseudo primitive idempotent of $\Gamma$, we mean a nonzero scalar multiple of the matrix ${\sum }_{i=0}^{D}{\sigma }_{i}{A}_{i}$, where ${A}_{0},{A}_{1},...,{A}_{D}$ are the distance matrices of $\Gamma$. Given pseudo primitive idempotents $E,F$ of $\Gamma$, 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 $\Gamma$. In this paper, we determine all the taut pairs of pseudo primitive idempotents of $\Gamma$.

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/

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