26 research outputs found
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
The chromatic number of the q-Kneser graph for large q
We obtain a new weak Hilton-Milner type result for intersecting families ofk-spaces inF2kq, which improves several known results. In particular the chromaticnumber of theq-Kneser graphqKn:kwas previously known forn >2k(except forn= 2k+1 andq= 2) ork 5, so that the only remaining open cases are (n, k) = (2k, k) withq∈{2,3,4}and (n, k) = (2k+ 1, k) withq= 2
Intriguing sets of partial quadrangles
The point-line geometry known as a \textit{partial quadrangle} (introduced by
Cameron in 1975) has the property that for every point/line non-incident pair
, there is at most one line through concurrent with . So in
particular, the well-studied objects known as \textit{generalised quadrangles}
are each partial quadrangles. An \textit{intriguing set} of a generalised
quadrangle is a set of points which induces an equitable partition of size two
of the underlying strongly regular graph. We extend the theory of intriguing
sets of generalised quadrangles by Bamberg, Law and Penttila to partial
quadrangles, which surprisingly gives insight into the structure of hemisystems
and other intriguing sets of generalised quadrangles
Incidence geometry from an algebraic graph theory point of view
The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense.
The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte.
Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them.
Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight.
A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5.
Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems.
The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch
Cameron-Liebler -sets in
We study Cameron-Liebler -sets in the affine geometry, so sets of
-spaces in . This generalizes research on Cameron-Liebler
-sets in the projective geometry . Note that in algebraic
combinatorics, Cameron-Liebler -sets of correspond to
certain equitable bipartitions of the Association scheme of -spaces in
, while in the analysis of Boolean functions, they correspond
to Boolean degree functions of . We define Cameron-Liebler
-sets in by intersection properties with -spreads and
show the equivalence of several definitions. In particular, we investigate the
relationship between Cameron-Liebler -sets in and
. As a by-product, we calculate the character table of the
association scheme of affine lines. Furthermore, we characterize the smallest
examples of Cameron-Liebler -sets. This paper focuses on
for , while the case for Cameron-Liebler line classes in was already treated separately
Cameron-Liebler sets of k-spaces in PG(n,q)
Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F.
Ihringer. We list several equivalent definitions for these Cameron-Liebler
sets, by making a generalization of known results about Cameron-Liebler line
sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also
present a classification result
On the incidence map of incidence structures
By using elementary linear algebra methods we exploit properties of the incidence map of certain incidence structures with finite block sizes. We give new and simple proofs of theorems of Kantor and Lehrer, and their infinitary version. Similar results are obtained also for diagrams geometries.
By mean of an extension of Block’s Lemma on the number of orbits of an automorphism group of an incidence structure, we give informations on the number of orbits of: a permutation group (of possible infinite degree) on subsets of finite size; a collineation group of a projective and affine space (of possible infinite dimension) over a finite field on subspaces of finite dimension; a group of isometries of a classical polar space (of possible infinite rank) over a finite field on totally isotropic subspaces (or singular in case of orthogonal spaces) of finite dimension.
Furthermore, when the structure is finite and the associated incidence matrix has full rank, we give an alternative proof of a result of Camina and Siemons. We then deduce that certain families of incidence structures have no sharply transitive sets of automorphisms acting on blocks
Cameron-Liebler sets of k-spaces in PG(n,q)
Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F.
Ihringer. We list several equivalent definitions for these Cameron-Liebler
sets, by making a generalization of known results about Cameron-Liebler line
sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also
present a classification result