Large {0,1,…,t}-cliques in dual polar graphs

Abstract

We investigate {0,1,…,t}-cliques of generators on dual polar graphs of finite classical polar spaces of rank d. These cliques are also known as Erdős–Ko–Rado sets in polar spaces of generators with pairwise intersections in at most codimension t. Our main result is that we classify all such cliques of maximum size for t≤8d/5−2 if q≥3, and t≤8d/9−2 if q=2. We have the following byproducts.(a)For q≥3 we provide estimates of Hoffman's bound on these {0,1,…,t}-cliques for all t.(b)For q≥3 we determine the largest, second largest, and smallest eigenvalue of the graphs which have the generators of a polar space as vertices and where two generators are adjacent if and only if they meet in codimension at least t+1. Furthermore, we provide nice explicit formulas for all eigenvalues of these graphs.(c)We provide upper bounds on the size of the second largest maximal {0,1,…,t}-cliques for some t.