Completely regular clique graphs. II

Suzuki, Hiroshi

Journal of Algebraic Combinatorics, Tome 43 (2016) no. 2, pp. 417-445.

Résumé

Let $\varGamma = (X,R)$ be a connected graph. Then $\varGamma $ is said to be a completely regular clique graph of parameters $(s,c)$ with $s\ge 1$ and $c\ge 1$, if there is a collection $\mathcal {C}$ of completely regular cliques of size $s+1$ such that every edge is contained in exactly $c$ members of $\mathcal {C}$. In the previous paper [the author, ibid. 40, No. 1, 233--244 (2014; Zbl 1297.05268)], we showed, among other things, that a completely regular clique graph is distance-regular if and only if it is a bipartite half of a certain distance-semiregular graph. In this paper, we show that a completely regular clique graph with respect to $\mathcal {C}$ is distance-regular if and only if every $\mathcal {T}(C)$-module of endpoint zero is thin for all $C \in \mathcal {C}$. We also discuss the relation between a $\mathcal {T}(C)$-module of endpoint 0 and a $\mathcal {T}(x)$-module of endpoint 1 and study examples of completely regular clique graphs.

Détail

Classification : 05C69, 05C40, 05C30, 05C12
Keywords: distance-regular graph, association scheme, subconstituent algebra, Terwilliger algebra, completely regular code, distance-semiregular graph