Completely regular clique graphs,II

Let (varGamma = (X,R)) be a connected graph. Then (varGamma ) is said to be a completely regular clique graph of parameters (s, c) with (sge 1) and (cge 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 (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 ({mathcal {C}}) is distance-regular if and only if every ({mathcal {T}}(C))-module of endpoint zero is thin for all (Cin {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.