Distance-regular graphs of large diameter that are completely regular clique graphs

A connected graph is said to be a completely regular clique graph with parameters (s, c), (s, c in {mathbb {N}}), 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}). It is known that many families of distance-regular graphs are completely regular clique graphs. In this paper, we determine completely regular clique graph structures, i.e., the choices of (mathcal {C}), of all known families of distance-regular graphs with unbounded diameter. In particular, we show that all distance-regular graphs in this category are completely regular clique graphs except the Doob graphs, the twisted Grassmann graphs and the Hermitean forms graphs. We also determine parameters (s, c); however, in a few cases we determine only s and give a bound on the value c. Our result is a generalization of a series of works by J. Hemmeter and others who determined distance-regular graphs in this category that are bipartite halves of bipartite distance-regular graphs.