Large $\{0,1,\dots ,t\}$-Cliques in Dual Polar Graphs

Abstract

We investigate $\{0,1,\dots ,t\}$-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 $t\le \sqrt{8d/5}-2$ if $q\ge 3$, and $t\le \sqrt{8d/9}-2$ if $q=2$. We have the following byproducts. (a) For $q\ge 3$ we provide estimates of Hoffman's bound on these $\{0,1,\dots ,t\}$-cliques for all $t$. (b) For $q\ge 3$ 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 $\{0,1,\dots ,t\}$-cliques for some $t$.

arXiv comment30 pages including appendix