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Madžarska raziskovalna skupina se večinoma nahaja na
ELTE in MTA-ELTE raziskovalni skupini za geometrijsko in algebraično
kombinatoriko, Slovenska raziskovalna skupina pa se večinoma nahaja na
Univerzi na Primorskem. Sodelovanje med slovensko in madžarsko raziskovalno
skupino se je začelo konec devetdesetih let. Od takrat je bilo več obiskov v obeh smereh v okviru slovensko-madžarskih
medvladnih znanstvenih in tehnoloških projektov. Poudarek je bil vedno na
algebraičnih metodah, ki se uporabljajo pri teoriji grafov in (končni)
geometriji. Od leta 2015 sodelujemo na podobnih raziskovalnih temah v okviru
OTKA-ARRS projekta. To sodelovanje je privedlo do več skupnih publikacij. Imeli
smo plodne diskusije tudi o številnih drugih raziskovalnih problemih. Sedanji
predlog je naravno nadaljevanje našega skupnega dela. Dve raziskovalni
skupini sodelujeta tudi pri poučevanju: Univerza na Primorskem in Eotvos
Lorand Univerza imata aktiven Erasmus sporazum. Poleg osmih madžarskih in devetih
slovenskih kolegov, načrtujemo tudi tesno sodelovanje z Gaborjem Korchmarosem
(Potenza, Italija), ki ima globoko znanje iz teorije grup in geometrije ter o
njihovi uporabi v teoriji grafov. Še posebej, napisal je več člankov o grupah
avtomobilizmov grafov in geometrij.
V tem projektu želimo nadaljevati in okrepiti naše
sodelovanje pri uporabi algebraičnih,
geometrijskih in kombinatoričnih metod za probleme o grafih, grupah,
konfiguracijah in geometrijah. V nekaterih primerih so aktualne raziskovalne
teme izhajale iz našega dela v prejšnjem OTKA-ARRS projektu. Splošne teme in
cilji se niso spremenili, želeli bi delati na problemih, ki razkrivajo
povezave med geometrijo, grafi in grupami. V nekem smislu je to tudi cilj
slovenskih revij Ars Mathematica Contemporanea in Art of Discrete and Applied
Mathematics. Seveda so konkretni problemi v tej prijavi različni. Ta projekt
bomo uporabili za podporo pri izmenjavi doktorskih študentov, in
sicer, načrtujemo organizacijo delavnic za širjenje rezultatov, ki jih bomo pridobili v okvirju tega
projekta.
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V okviru
tega projekta želimo delati na naslednjih raziskovanih temah: Konfiguracije,
grafi in grupe iz kvadratnih form nad končnimi polji, razdaljno regularni
grafi, povezavno ožinsko regularni grafi, Frobeniusove grafične in digrafične
predstavitve, CI lastnosti grafov in geometrij, jedra in končne geometrije.
Treba je opozoriti, da spodaj navedeni seznam ni nujno popoln seznam
raziskovalnih problemov, na katerih želimo delati v okviru tega projekta. Med
našimi raziskavami se lahko pojavijo tudi druga odprta vprašanja, povezana z
grafi, grupami, konfiguracijami in geometrijami.
ANG
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The
Hungarian research group is mostly located at ELTE and the MTA-ELTE Geometric
and Algebraic Combinatorics Research Group and the Slovenian research group
is mostly located at University of Primorska. The cooperation of this
research group with our Slovenian
colleagues has started in the end of 1990s. Since then several visits have
taken place in both directions in the framework of Slovenian-Hungarian Intergovernmental
Scientific and Technological Projects. The focus was always on algebraic
methods applied to graph theory and (finite) geometry. Since 2015 we have
been cooperating on similar topics in the framework of an OTKA-ARRS project.
This collaboration led to several joint papers and we had fruitful
discussions about many other papers. The present proposal is natural
continuation of our joint work. We also have a cooperation in teaching:
University of Primorska and Eötvös Loránd University have an active Erasmus
agreement. Besides the eight Hungarian and nine Slovenian colleagues, we also
plan to work closely with Gábor Korchmáros (Potenza, Italy), who has a deep
knowledge in group theory and geometry, and their application to graph
theory. In particular, he wrote several papers on automorhism groups of
graphs and geometries.
In
the present project we would like to continue and strengthen our cooperation
in the use of algebraic, geometric, and combinatorial techniques for problems
about graphs, groups, configurations
and geometries. In some cases the current topics grew out of our work in the
previous OTKA-ARRS project. From a distance, the topics and the aims did not
change, we would like to attack problems which reveal connections between
geometris, graphs and groups. In a sense, this is also the aim of the
Slovenian journals Ars Math. Contemporanea and Art of Discrete and Applied
Maths. Of course, the concrete problems are different. We plan to use this
project to support exchange of Ph.D. students, in particular, we plan to
organize workshops to spread the results obtained in the project.
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We have
the following specific topics on which we plan to work on: Configurations;
Graphs and groups from quadratic forms over finite fields; Distance-regular Cayley
graphs; Edge-girth regular graphs; Frobenius graphical and digraphical
representations; CI-property of graphs and geometries; Cores and finite geometry.
It should be noted that the above list is necessarily not the full list of problems
that we wish to attack together. During our research (many) other problems
related to graphs, groups, configurations and geometries may occur.
