Buildings and Kneser graphs

Abstract

If C is a collection of mutually intersecting k-subsets of a fixed n-set, how big can C be? And in the extreme case, what is the structure of C? This question was answered by Erd?os, Ko, and Rado, (already in 1938, but they first published this result in 1961): One must have ... , and if equality holds then C is the collection of all k-subsets containing some fixed element of the given n-set. The Kneser graph K(n, k) is the graph with as vertices the k-subsets of a fixed n-set, where two k-subsets are adjacent when they are disjoint. In this terminology, Erdos, Ko, and Rado, found the largest cocliques (independent sets of vertices) in K(n, k). Many people have studied generalizations and variations of this problem, and that is also what we do in this thesis. We have objects and some kind of a distance function, and define a Kneser graph with our objects as vertices, two objects being adjacent when they are far apart, have maximum distance. The goal is always to find the maximum size of a coclique in such a graph, and to characterize the cases that reach this maximum. Our objects will usually be flags of some fixed type in a finite building. Relations between flags are parameterized by double cosets in the Weyl group W, and being far apart can be defined as having the relation corresponding to the double coset containing the longest word w0. In simple cases we succeed in classifying all maximal cocliques in our Kneser graphs (not only the ones of maximal size). See e.g. [11], where the conjecture from [82] is proved. It turns out that most of the larger examples can be derived by a simple construction from corresponding examples in an apartment of the building, [22]. In not-so-simple examples we sometimes succeed in determining the maximum coclique size, and sometimes only obtain bounds. One way to obtain bounds is via Delsarte’s LP bound, and we have to compute the eigenvalues of our Kneser graphs. These eigenvalues turn out to be prime powers, and that leads to results on p-rank and Smith normal form of the adjacency matrices.