(SI) Cilj predlaganega projekta je raziskati številne nove možne pristope v smeri dokaza Weissove domneve. Prvi pristop je najbližji klasičnim metodam, vendar nameravamo v njem uporabiti nedavno dokazane in globoke rezultate s področja lokalne analize in končnih grup. Drugi pristop je popolnoma nov, osnovan je na pozitivni rešitvi omejenega Burnsideovega problema, ki nam omogoča prevesti Weissovo domnevo na dva ločena problema, o eksponentu in rangu vozliščnega stabilizatorja, kadar je lokalno delovanje primitivno. Tretji pristop, ki ga želimo raziskati, je posplošitev (Weissove domneve) na splošnejše vprašanje o (hitrosti) rasti vozliščnega stabilizatorja glede na red grafa (v primerjavi z redom grafa). Čeprav se dobro zavedamo, da je dokazati 30 let staro domnevo ambiciozen cilj, ki ga morda ne bomo dosegli v času izvajanja predlaganega raziskovalnega projekta, verjamemo, da nam bodo orodja, ki so nam nedavno postala dostopna, pa tudi odlična raziskovalna skupina, omogočili najmanj to, da dosežemo velik napredek v smeri končne razrešitve domneve, obenem pa izboljšamo naše razumevanje simetrij pri grafih.

(EN) The aim of the proposed project is to explore several new possible approaches towards the proof the Weiss conjecture. The first approach is closest to the classical methods but plans to utilise recently proved and deep results in the area of local analysis of finite groups. The second approach is completely new and is based on the positive solution of the restricted Burnside problem, which allows one to reduce the Weiss conjecture to two separate problems about the exponent and rank of the vertex-stabiliser when the local action thereof is primiitve. The third approach that we want to explore is by generalising to a more general question about the growth of the vertex-stabiliser with respect to the order of the graph. While we fully acknowledge that proving this 30 years old conjecture is an ambitious goal which might not be achieved during the duration of the proposed research project, we believe that the tools that have recently become available to us and an excellent research team will at least allow us to make a considerable progress towards the final resolution of the conjecture and at the same time increase our understanding of the symmetry in graphs.