SL(2,Z) Representations and 2-Semiregular Modular Categories

We address the open question of which representations of the modular group SL(2,Z) can be realized by a modular category. In order to investigate this problem, we introduce the concept of a symmetrizable representation of SL(2,Z) and show that this property is necessary for the representation to be realized. We then prove that all congruence representations of SL(2,Z) are symmetrizable. The proof involves constructing a symmetric basis, which greatly aids in further calculation. We apply this result to the reconstruction of modular category data from representations, as well as to the classification of semiregular categories, which are defined via an action of the Galois group Gal(Qbar/Q) on their simple objects

Distance-Biregular Graphs and Orthogonal Polynomials

This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems. We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound

Maximal arcs and group divisible designs

The existence of maximal arcs of a certain type in symmetric designs is shown to yield semiregular group divisible designs whose duals are also semiregular group divisible. Two infinite families of such group divisible designs are constructed. The group divisible designs in these families are, in general, not symmetric

A combinatorial approach to optimal designs.

PhDA typical problem in experimental design theory is to find a block design in a class that is optimal with respect to some criteria, which are usually convex functions of the Laplacian eigenvalues. Although this question has a statistical background, there are overlaps with graph and design theory: some of the optimality criteria correspond to graph properties and designs considered ‘nice’ by combinatorialists are often optimal. In this thesis we investigate this connection from a combinatorial point of view. We extend a result on optimality of some generalized polygons, in particular the generalized hexagon and octagon, to a third optimality criterion. The E-criterion is equivalent with the graph theoretical problem of maximizing the algebraic connectivity. We give a new upper bound for regular graphs and characterize a class of E-optimal regular graph designs (RGDs). We then study generalized hexagons as block designs and prove some properties of the eigenvalues of the designs in that class. Proceeding to higher-dimensional geometries, we look at projective spaces and find optimal designs among two-dimensional substructures. Some new properties of Grassmann graphs are proved. Stepping away from the background of geometries, we study graphs obtained from optimal graphs by deleting one or several edges. This chapter highlights the currently available methods to compare graphs on the A- and D-criteria. The last chapter is devoted to designs to which a number of blocks are added. Cheng showed that RGDs are A- and D-optimal if the number of blocks is large enough for which we give a bound and characterize the best RGDs in terms of their underlying graphs. We then present the results of an exhaustive computer search for optimal RGDs for up to 18 points. The search produced examples supporting several open conjectures

Some classes of spectrally constrained graphs

Spektralna teorija grafova je grana matematike koja je nastala pedesetih godina pro²log veka i od tada se neprestano razvija. Njen zna£aj ogleda se u brojnim primenama, naro£ito u hemiji, zici, ra£unarstvu i drugim naukama. Grane matematike, kao ²to su linearna algebra i, posebno, teorija matrica imaju vaºnu ulogu u spektralnoj teoriji grafova. Postoje razli£ite matri£ne reprezentacije grafa. Najvi ²e su izu£avane matrica susedstva grafa i Laplasova (P.S. Laplace) matrica, a zatim i Zajdelova (J.J. Seidel) i takozvana nenegativna Laplasova matrica. Spektralna teorija grafova u su²tini uspostavlja vezu izme u strukturalnih osobina grafa i algebarskih osobina njegove matrice, odnosno razmatra o kojim se strukturalnim osobinama (kao ²to su povezanost, bipartitnost, regularnost i druge) mogu dobiti informacije na osnovu nekih svojstava sopstvenih vrednosti njegove matrice. Veliki broj dosada²njih rezultata iz ovog ²irokog polja istraºivanja moºe se na¢i u slede¢im monograjama: [20], [21], [23] i [58]. Disertacija sadrºi originalne rezultate dobijene u nekoliko podoblasti spektralne teorije grafova. Ti rezultati izloºeni su u tri celine glave, od kojih je svaka podeljena na poglavlja, a neka od njih na potpoglavlja. Na po£etku svake glave, u posebnom poglavlju, formulisan je problem koji se u toj glavi razmatra, kao i postoje¢i rezultati koji se odnose na zadati problem, a neophodni su za dalja razmatranja. U ostalim poglavljima predstavljeni su originalni rezultati, koji se nalaze i u radovima [3], [4], [47], [48], [49], [50], [51] i [52]. U prvoj glavi razmatra se druga sopstvena vrednost regularnih grafova. Postoji dosta rezultata o grafovima £ija je druga po veli£ini sopstvena vrednost ograni£ena odozgo nekom (relativno malom) konstantom. Posebno, druga sopstvena vrednost ima zna£ajnu ulogu u odre ivanju strukture regularnih grafova. Poznata je karakterizacija regularnih grafova koji imaju samo jednu pozitivnu sopstvenu vrednost (videti [20]), a razmatrani su i regularni grafovi sa osobinom 2 ≤ 1 (videti [64]). U okviru ove disertacije pro²iruju se rezultati koji se nalaze u radu [64], a predstavljaju se i neki op²ti rezultati koji se odnose na vezu odre enih spektralnih i strukturalnih osobina regularnih nebipartitnih grafova bez trouglova...Spectral graph theory is a branch of mathematics that emerged more than sixty years ago, and since then has been continuously developing. Its importance is reected in many interesting and remarkable applications, esspecially in chemistry, physics, computer sciences and other. Other areas of mathematics, like linear algebra and matrix theory have an important role in spectral graph theory. There are many dierent matrix representations of a given graph. The ones that have been studied the most are the adjacency matrix and the Laplace matrix, but also the Seidel matrix and the so-called signless Laplace matrix. Basically, the spectral graph theory establishes the connection between some structrural properties of a graph and the algebraic properties of its matrix, and considers structural properties that can be described using the properties of the eigenvalues of its matrix. Systematized former results from this vast eld of algebraic graph theory can be found in the following monographs: [20], [21], [23] i [58]. This thesis contains original results obtained in several subelds of the spectral graph theory. Those results are presented within three chapters. Each chapter is divided into sections, and some sections into subsections. At the beginning of each chapter (in an appropriate sections), we formulate the problem considered within it, and present the existing results related to this problem, that are necessary for further considerations. All other sections contain only original results. Those results can also be found in the following papers: [3], [4], [47], [48], [49], [50], [51] and [52]. In the rst chapter we consider the second largest eigenvalue of a regular graph. There are many results concerning graphs whose second largest eigenvalue is upper bounded by some (relatively small) constant. The second largest eigenvalue plays an important role in determining the structure of regular graphs. There is a known characterization of regular graphs with only one positive eigenvalue (see [20]), and regular graphs with the property 2 ≤ 1 have also been considered (see [64]). Within this thesis we extend the results given in [64], and we also present some general results concerning the relations between some structural and spectral properties of regular triangle-free graphs..