A connected distance-regular graph on two or more vertices that contains
a set of Delsarte cliques such that every edge of lies in a unique member of is known as a geometric distance-regular
graph (Koolen *et al. *2023). The concept of geometric graphs was first
defined by Bose (1963) for strongly regular
graphs and subsequently extended to distance-regular
graphs by Godsil (1993).

Koolen *et al. *(2023) enumerate 18 cases of non-geometric distance-regular graphs of graph diameter at least 3 with smallest
graph eigenvalue at least , where the distance-regular
graph with intersection array is putative and the odd -cycles with (which satisfy all the given criteria) are apparently
silently omitted.

Classes of graphs that are geometric include the Johnson graphs and Hamming graphs (Koolen *et al.
*2023), which in turn include complete graphs,
hypercube graphs, rook graphs, triangular
graphs, and tetrahedral Johnson graphs.