Given a connected distance-regular graph on two or more vertices with vertex degree 
and smallest graph
eigenvalue 
,
every clique satisfies the inequality
 |
known as the Delsarte bound (Koolen et al. 2023), where 
is the size of the clique and
is the absolute
value. This inequality was originally proved for strongly
regular graphs (Delsarte 1973) and subsequently generalized to distance-regular
graphs (Godsil 1993).
A clique for which the inequality becomes an equality is known as a Delsarte clique, and a distance-regular graph 
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).
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15 Nov 2023.