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

MacLean, Mark S.; Miklavič, Štefko

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

Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D\ge 4$, valency $k\ge 3$, and intersection numbers $c_i,b_i(0\le i\le D)$. 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 $\Gamma$.

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

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