14 research outputs found
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 â 1, q^2) with d odd, the bound is new and even tight.
A variant of the famous ErdĆs-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 â 1, q^2) for odd rank d â„ 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
Embedding dimensions of simplicial complexes on few vertices
We provide a simple characterization of simplicial complexes on few vertices
that embed into the -sphere. Namely, a simplicial complex on vertices
embeds into the -sphere if and only if its non-faces do not form an
intersecting family. As immediate consequences, we recover the classical van
Kampen--Flores theorem and provide a topological extension of the Erd\H
os--Ko--Rado theorem. By analogy with F\'ary's theorem for planar graphs, we
show in addition that such complexes satisfy the rigidity property that
continuous and linear embeddability are equivalent.Comment: 8 page
A collection of open problems in celebration of Imre Leader's 60th birthday
One of the great pleasures of working with Imre Leader is to experience his
infectious delight on encountering a compelling combinatorial problem. This
collection of open problems in combinatorics has been put together by a subset
of his former PhD students and students-of-students for the occasion of his
60th birthday. All of the contributors have been influenced (directly or
indirectly) by Imre: his personality, enthusiasm and his approach to
mathematics. The problems included cover many of the areas of combinatorial
mathematics that Imre is most associated with: including extremal problems on
graphs, set systems and permutations, and Ramsey theory. This is a personal
selection of problems which we find intriguing and deserving of being better
known. It is not intended to be systematic, or to consist of the most
significant or difficult questions in any area. Rather, our main aim is to
celebrate Imre and his mathematics and to hope that these problems will make
him smile. We also hope this collection will be a useful resource for
researchers in combinatorics and will stimulate some enjoyable collaborations
and beautiful mathematics
Global hypercontractivity and its applications
The hypercontractive inequality on the discrete cube plays a crucial role in
many fundamental results in the Analysis of Boolean functions, such as the KKL
theorem, Friedgut's junta theorem and the invariance principle. In these
results the cube is equipped with the uniform measure, but it is desirable,
particularly for applications to the theory of sharp thresholds, to also obtain
such results for general -biased measures. However, simple examples show
that when , there is no hypercontractive inequality that is strong
enough.
In this paper, we establish an effective hypercontractive inequality for
general that applies to `global functions', i.e. functions that are not
significantly affected by a restriction of a small set of coordinates. This
class of functions appears naturally, e.g. in Bourgain's sharp threshold
theorem, which states that such functions exhibit a sharp threshold. We
demonstrate the power of our tool by strengthening Bourgain's theorem, thereby
making progress on a conjecture of Kahn and Kalai and by establishing a
-biased analog of the invariance principle.
Our results have significant applications in Extremal Combinatorics. Here we
obtain new results on the Tur\'an number of any bounded degree uniform
hypergraph obtained as the expansion of a hypergraph of bounded uniformity.
These are asymptotically sharp over an essentially optimal regime for both the
uniformity and the number of edges and solve a number of open problems in the
area. In particular, we give general conditions under which the crosscut
parameter asymptotically determines the Tur\'an number, answering a question of
Mubayi and Verstra\"ete. We also apply the Junta Method to refine our
asymptotic results and obtain several exact results, including proofs of the
Huang--Loh--Sudakov conjecture on cross matchings and the
F\"uredi--Jiang--Seiver conjecture on path expansions.Comment: Subsumes arXiv:1906.0556
A mathematical foundation for the use of cliques in the exploration of data with navigation graphs
Navigation graphs were introduced by Hurley and Oldford (2011) as a graph-theoretic framework for exploring data sets, particularly those with many variables. They allow the user to visualize one small subset of the variables and then proceed to another subset, which shares a few of the original variables, via a smooth transition. These graphs serve as both a high level overview of the dataset as well as a tool for a first-hand exploration of regions deemed interesting.
This work examines the nature of cliques in navigation graphs, both in terms of type and magnitude, and speculates as to what their significance to the underlying dataset might be. The questions answered by this body of work were motivated by the belief that the presence of cliques in navigation graphs is a potential indicator for the existence of an interesting, possibly unanticipated, relationship among some of the variables.
In this thesis we provide a detailed examination of cliques in navigation graphs, both in terms of type, size and number. The study of types of cliques informs us of the potential significance of highly connected structures to the underlying data and guides our approach for examining the possible clique sizes and counts. On the other hand, the prevalence of large clique sizes and counts is suggestive of an interesting, possibly unexpected, relationship between the variates in the data.
To address the challenges surrounding the nature of cliques in navigation graphs, we develop a framework for the derivation of closed-form expressions for the moments of count random variables in terms of their underlying indecomposable summands is established. We use this framework in conjunction with a connection between intersecting set families to obtain edge counts within a clique cover and thus, obtain closed-form expressions for the moments of clique counts in random graphs
State Transfer & Strong Cospectrality in Cayley Graphs
This thesis is a study of two graph properties that arise from quantum walks: strong cospectrality of vertices and perfect state transfer. We prove various results about these properties in Cayley graphs.
We consider how big a set of pairwise strongly cospectral vertices can be in a graph. We prove an upper bound on the size of such a set in normal Cayley graphs in terms of the multiplicities of the eigenvalues of the graph. We then use this to prove an explicit bound in cubelike graphs and more generally, Cayley graphs of . We further provide an infinite family of examples of cubelike graphs (Cayley graphs of ) in which this set has size at least four, covering all possible values of .
We then look at perfect state transfer in Cayley graphs of abelian groups having a cyclic Sylow-2-subgroup. Given such a group, G, we provide a complete characterization of connection sets C such that the corresponding Cayley graph for G admits perfect state transfer. This is a generalization of a theorem of Ba\v{s}i\'{c} from 2013, where he proved a similar characterization for
Cayley graphs of cyclic groups
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum