Grafi in lihi avtomorfizmi / Graphs and Odd Automorphisms
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Naziv Tittle |
Grafi in lihi avtomorfizmi / Graphs and Odd Automorphisms |
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Akronim Acronim |
N1-0038 |
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Opis Description |
(SI) Projekt obravnava koncept sodih/lihih avtomorfizmov grafov, t. j. sode/lihe permutacije na množici točk grafa, ki ohranjajo strukturo grafa. Ta preprost, vendar nov koncept odpira novo raziskovalno področje, ki je zanimivo že samo po sebi in vodi do pomembnih novih dognanj pri mnogih že vrsto let odprtih raziskovalnih problemih.
Glavni raziskovalni problemi so: Za dano tranzitivno grupo sodih avtomorfizvom H grafa X ugotovi, ali obstaja grupa G, ki vsebuje lihe avtomorfizme grafa X in grupo H kot podgrupo. Če je odgovor pritrdilen, določi minimalni indeks [G:H] med vsemi takimi grupami G. Če grupa H premore tako H-invariantno particijo B, da je delovanje grupe H na kvocientu X/B primitivno, ugotovi, ali obstaja takšna "ekstenzija" G, da je indeks [G:H] minimalen in B ni G-invariantna particija.
Iskanje "ekstenzij" grup skozi lihe avtomorfizme je tudi nov pristop k reševanju naslednjega temeljnega vprašanja: Za dano tranzitivno grupo H, ki deluje na kombinatoričnem / geometrijskem objektu, ugotovi, ali je H cela grupa avtomorfizmov danega objekta ali ne. Kadar H sestoji le iz sodih avtomorfizmov, lahko na to vprašanje podamo delni odgovor, v kolikor struktura danega objekta zahteva obstoj lihih avtomorfizmov.
Cilj projekta je zgraditi teorijo, ki bo omogočala, da za dani graf, ki premore tranzitivno grupno delovanje, ugotovimo, ali premore ali ne premore lihih avtomorfizmov. (EN) The project deals with the concept of even/odd automorphisms of graphs, i.e. even/odd permutations on the set of vertices of a graph that preserve the structure of the graph. This simple but novel concept opens up a new area of research, which is interesting in itself and leads to important new insights into many research problems that have been open for many years. The main research problems are: For a given transitive group of even automorphisms H of a graph X, determine whether there exists a group G containing the odd automorphisms of X and the group H as a subgroup. If so, determine the minimal index [G:H] among all such groups G. If group H has an H-invariant partition B such that the action of group H on the quotient X/B is primitive, determine whether there exists an "extension" of G such that the index [G:H] is minimal and B is not a G-invariant partition. Finding "extensions" of groups via odd automorphisms is also a new approach to solving the following fundamental question: for a given transitive group H acting on a combinatorial/geometric object, determine whether or not H is the entire group of automorphisms of the given object. When H consists of only even automorphisms, we can give a partial answer to this question if the structure of the given object requires the existence of odd automorphisms. The goal of the project is to construct a theory that will allow us to determine, for a given graph that admits a transitive group action, whether or not it admits odd automorphisms. |
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Vrsta projekta Project Type |
Temeljni projekt |
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Trajanje Duration |
01/09/2015 - 31/08/2018 |
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URL URL |
https://www.iam.upr.si/sl/oddelki/ma/Projekti/N1-0038/ |
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Vodja projekta Project Leader |
Dragan Marušič |
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Sodelujoče organizacije Participating organizations |
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Oddelek Department |
Oddelek za matematiko IAM |