V okviru predlaganega projekta se sprašujemo po obstoju polregularnih avtomorfizmov, ki imajo dodatno lastnost, t.j. da pripadajoči kvocientni grafiohranijo valenco začetnega grafa. Bolj natančno, zahtevamo, da so pripadajoči kvocientni »multigrafi« glede na orbite podgrupe generirane spolregularnim avtomorfizmom pravzaprav enostavni grafi. Avtomorfizmu, ki zadosti temu pogoju, bomo rekli simplicialen avtomorfizem. Ciljpredlaganega projekta je pričetek raziskovalnega dela, v okviru katerega se bo določilo, kateri točkovno tranzitivni grafi premorejo simplicialneavtomorfizme. V okviru predlaganega projekta bomo pričeli to raziskovalno delo z obravnavo razreda kubičnih točkovno tranzitivnih grafov.Natančneje, cilj predlaganega projekta je podati odgovor na vprašanje, kateri kubični točkovno tranzitivni grafi s kvaziprimitivno grupoavtomorfizmov premorejo simplicialne avtomorfizme. Obstoj simplicialnih avtomorfizmov bo obravnavan tudi za druge razrede grafov. Poleg tegabo koncept simplicialnih avtomorfizmov uporabljen za reševanje različnih odprtih problemov v algebraični teoriji grafov.
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Within the proposed project we ask for existence of semiregular automorphisms having the additional property that the corresponding quotientgraphs preserve the valencies of the original graphs. More precisely, we require that the quotient ``multigraphs'' with respect to the orbits of asubgroup generated by a semiregular automorphism are in fact simple graphs. An automorphism satisfying this condition will be called simplicial.The aim of this proposal is to launch the project of determining which vertex-transitive graphs admit simplicial automorphisms by starting theanalysis for the class of cubic vertex-transitive graphs. In particular, the proposed project aims to give a complete answer to the question whichcubic vertex-transitive graph with a quasiprimitive automorphism group admit simplicial automorphisms of prime order. Existence of simplicalautomorphisms in other classes of graphs will also be considered. In addition, the concept of simplical automorphism will be applied to differentopen problems in algebraic graph theory.
Given a permutation group , a subset I of G is called intersecting if, for any two permutations g and h in I, there exists v in V such that g(v)=h(v). Apermutation group G is said to have the Erdos-Ko-Rado property (in short EKR-property), if the size of a maximum intersecting set is equal to theorder of the largest point stabilizer. The measure of how far a transitive permutation group G is from having the EKR-property is the so-calledintersection density r(G) defined as the quotient r(G)=|I|/Gv, where I is an intersecting set in G of a maximum size, and Gv is a point stabilizer of G.
