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

摘要

A connected graph is said to be a completely regular clique graph with parameters (s,c), s,c is an element of N, 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. 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 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.