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

TL;DR

This paper classifies maximum size $ ext{0, 1, ..., t}$-cliques in dual polar graphs of finite classical polar spaces, extending understanding of Erdős-Ko-Rado sets and eigenvalue bounds in algebraic combinatorics.

Contribution

It provides a classification of maximum size $ ext{0, 1, ..., t}$-cliques for certain parameters, along with eigenvalue formulas and bounds, advancing combinatorial and spectral analysis of dual polar graphs.

Findings

Classified maximum size $ ext{0, 1, ..., t}$-cliques for specific $t$ and $q$.

Derived explicit eigenvalue formulas for associated graphs.

Provided bounds on sizes of second largest maximal $ ext{0, 1, ..., t}$-cliques.

Abstract

We investigate $\{0, 1, \ldots, t \}$-cliques of generators on dual polar graphs of finite classical polar spaces of rank $d$. These cliques are also known as Erd\H{o}s-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 $t \leq \sqrt{8d/5}-2$ if $q \geq 3$, and $t \leq \sqrt{8d/9}-2$ if $q = 2$. We have the following byproducts. (a) For $q \geq 3$ we provide estimates of Hoffman's bound on these $\{0, 1, \ldots, t \}$-cliques for all $t$. (b) For $q \geq 3$ 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 $\{0, 1, \ldots, t \}$-cliques for some $t$.