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.
." 15 Nov 2023.