SLO

V tem projektu želimo še nadgraditi večletno izjemno plodno sodelovanje med slovenskimi in madžarskimi raziskovalci na področju algebraičnekombinatorike. Slovenski del projektne skupine prihaja iz Univerze na Primorskem, madžarski del pa predvsem iz Univerze v Szededu in iz UniverzeEotvos Lorand. Medtem ko je slovenski del projektne skupine v samem svetovnem vrhu na področju raziskav CI-grup, permutacijskih grup terdelovanja grup na grafih, pa je madžarski del skupine v samem svetovnem vrhu raziskav na področju končnih geometrij ter geometričnihkonfiguracij. Prav ta komplementarnost je porok, da bo sodelovanje obeh projektnih skupin izjemno uspešno.

V projektu se bomo ukvarjali predvsem z naslednjimi tematikami:

Klasifikacija nekaterih kombinatoričnih objektov, kot so "Bol loops" in "Sidonove množice". Kvazigrupa, ki premore identiteto, se imenuje "loop". Čeelementi takšne kvazigrupe zadoščajo identiteti x(y(xz)) = (x(yx))z, potem jo imenujemo (desna) "Bol loop". V tem projektu bomo predvsem študirali"Bol loops" reda 3.2^n. Množica S celih števil je Sidonova množica, če a+b = c+d (a,b,c,d ∈ S) implicira {a,b} = {c,d}. Definicija se da smiselnoposplošiti na poljubno abelsko grupo. V tem projektu bomo predvsem študirali Sidonove množice v vektorskih prostorih nad končnim poljem reda 2.

Nelinearnost vektorskih Boolovih funkcij. V tem projektu bi radi naredili znaten napredek pri razumevanju vektorske nelinearnosti vektorskih "almostperfect nonlinear" funkcij.

CI-grupe. Za razred Cayley-evih kombinatoričnih objektov neke končne grupe G si lahko postavimo naslednje vprašanje: ali je res, da sta dva objektaiz tega razreda izomorfna natanko takrat, ko sta izomorfna preko nekega avtomorfizma grupe G? Ko je to res, rečemo, da ima grupa G CI-lastnost zadani razred kombinatoričnih objektov. V tem projektu bomo predvsem študirali CI-lastnost grup oblike Z_p^n, kjer je p praštevilo.

Faktorizacija grup. Glej originalno projektno prijavo v priponki.

Generalizacija nekaterih izrekov Carlitz-ovega tipa. Glej originalno projektno prijavo v priponki.

Izreki Erdos-Ko-Rado-vega tipa za permutacijske grupe. V tem projektu nameravamo študirati tako imenovano EKR-lastnost tranzitivnihpermutacijskih grup, ki imajo cikličen vozliščni stabilizator.

Problemi posplošenih kletk in povezavno ožinsko regularni grafi. V tem delu projekta bi radi nadaljevali z že začetim raziskovanjem tega razredagrafov. Predvsem bi se radi osredotočili na povezave med temi grafi in nekaterimi problemi v ekstremalni teoriji grafov. Načrtujemo, da bomokonstruirali nove primere teh grafov, ki bodo izhajali iz končnih geometrij.

Geometrične konfiguracije in hipergrafi. V tem delu projekta se bomo osredotočili na razred sebi-dualnih konfiguracij. Raziskovali bomo predvsemkako bogat je ta razred konfiguracij.

Predlagani projekt je v samem vrhu dandanašnjih raziskav v algebraični teoriji grafov ter njunih aplikacijah v drugih znanostih. Pomembnost našihraziskovalnih ciljev je vidna iz bibliografij članov projektne skupine, njihove citiranosti in iz njihovih številnih povezav z raziskovalci v tujini. Rezultatitega projekta bodo predstavljali velik doprinos tako k razvoju algebraične teorije grafov kot tudi sorodnih področji. Ta doprinos pa bo seveda daldodaten zagon slovenski šoli algebraične teorije grafov. Projekt bo pomagal vsem članom raziskovalne skupine, da ostanejo v stiku z trenutnovodilnimi trendi v matematičnem raziskovanju, posebno na področju algebraične teorije grafov. Cilj projekta je tudi najti smernice za nadaljnoraziskovanje. Ponavadi se v procesu dokazovanja izrekov.

EN

In this project, we want to continue with a long-standing and extremely fruitful cooperation between Slovenian and Hungarian researchers in the fieldof algebraic combinatorics. The Slovenian part of the project group comes from the University of Primorska, while the Hungarian part mainly comes from the University of Szeded and from the Eotvos Lorand University. While the Slovenian part of the project group is at the very top in the field ofresearch on CI-groups, permutation groups and the action of groups on graphs, the Hungarian part of the group is at the very top in research in thefield of finite geometries and geometric configurations. This complementarity is the guarantee that the cooperation of the two project groups will beextremely successful.

In the project, we will mainly deal with the following topics:

Classification of combinatorial objects: Bol loops and Sidon sets. Quasigroups with a unit element are called loops. Loops satisfying the identityx(y(xz)) = (x(yx))z are called (right) Bol loops. In this project we will mainly study Bol loops of order 3.2^n. A set S of integers is a Sidon set or a Sidonsequence, if a+b = c+d, a,b,c,d ∈ S imply {a,b} = {c,d}. The definition can be generalized to any abelian group. In this project we will mainly study Sidonsets in vector spaces over the finite field of order 2.

Nonlinearity of vectorial Boolean functions. In this project, we want to make significant progress by the computation of vectorial nonlinearity ofalmost perfect nonlinear functions.

CI-groups. Given a class of Cayley combinatorial objects of a finite group, one can ask whether it is true that two such objects are isomorphic if andonly if they are isomorphic via a group automorphism. When this is so, the group is said to have the CI-property for the given class of combinatorialobjects. In this project we will study CI-property of the groups of form Z_p^n, where p is a prime number.

Group factorization. See the attached original application form.

Generalizations of Carlitz type theorems. See the attached original application form.

Erdos-Ko-Rado-type theorems for transitive permutation groups. In this project we will study the so-called EKR-property of transitive permutationgroups with cyclic point stabilizers.

Generalized cage problems and edge-girth-regular graphs. We would like to continue our ongoing research on this new class of graphs. In particular,the investigation of its connections to various extremal graph theory problems, and the classification of egr-graphs. We are planning to constructnew examples

arising from finite or infinite geometries.

Geometric configurations and hypergraphs. We will focus on the class of self-dual configurations. Various levels of self-duality lead to new sporadicexamples, and, to infinite classes of configurations; for example, the famous Grünbaum–Rigby configuration which is perfectly self-reciprocal. Aninteresting problem is to explore how rich is the class of such configurations.

The proposed project stands at the cutting edge of today's research in Algebraic Graph Theory and in Theory of incident structures, as well asapplications of these theories in other sciences. The importance of our research goals can be seen from project team members’ bibliographies, theircitations, and numerous links with scientists around the world. The outcome of this project will have a great impact to the development of theAlgebraic Graph Theory, as well as to related fields, and will give a strong impetus to the Slovenian Algebraic Graph Theory School. Project wouldhelp project team members to keep in touch with trends in modern mathematics, especially in algebraic graph theory. The second goal of thisproject is to find directions for further research. Usually, in the process of proving theorems, many such directions arise. We hope and belive that thiswill be the case in our project.