Completely regular clique graphs, II

Abstract

Let Γ = (X, R) be a connected graph. Then Γ is said to be a completely regular clique graph of parameters (s, c) with s ≥ 1 and c ≥ 1, if there is a collection C of completely regular cliques of size s + 1 such that every edge is contained in exactly c members of C. In the previous paper (Suzuki in J Algebr Combin 40:233-244, 2014), 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 C is distance-regular if and only if every T (C)-module of endpoint zero is thin for all C ∈ C. We also discuss the relation between a T (C)-module of endpoint 0 and a T (x)-module of endpoint 1 and study examples of completely regular clique graphs.