519 KB65 KB200820252017Amarra, CarmenGiudici, MichaelPraeger, Cheryl E.številka:1letnik:13str. 137-165COBISSID:18199385ISSN:1855-3966URN:URN:NBN:SI:doc-137M8L1IenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaCayleyevi grafilinearne grupepermutacijske grupesimetrični grafiAffine primitive symmetric graphs of diameter two|Let ?$n$? be a positive integer, ?$q$? be a prime power, and ?$V$? be a vector space of dimension ?$n$? over ?$\mathbb{F}_{q}$?. Let ?$G := V \rtimes G_{0}$?, where ?$G_{0}$? is an irreducible subgroup of ?$GL(V)$? which is maximal by inclusion with respect to being intransitive on the set of nonzero vectors. We are interested in the class of all diameter two graphs ?$\Gamma$? that admit such a group ?$G$? as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which ?$G_{0}$? is a subgroup of either ?$\Gamma\mathrm{L}(n,q)$? or ?$\Gamma \mathrm{Sp}(n, q)$? and is maximal in one of the Aschbacher classes ?$\mathcal{C}_{i}$?, where ?$i \in \{2, 4, 5, 6, 7, 8\}$?. We are able to determine all graphs ?$\Gamma$? which arise from ?$G_{0}\leq \Gamma\mathrm{L}(n,q)$? with ?$i \in \{2, 4, 8\}$?, and from ?$G_{0} \leq \Gamma \mathrm{Sp}(n, q)$? with ?$i \in \{2, 8\}$?. For the remaining classes we give necessary conditions in order for ?$\Gamma$? to have diameter two, and in some special subcases determine all ?$G$?-symmetric diameter two graphsNaj bo ?$n$? pozitivno celo število, ?$q$? potenca praštevila in ?$V$? vektorski prostor dimenzije ?$n$? nad ?$\mathbb{F}_{q}$?. Naj bo ?$G := V \rtimes G_{0}$?, kjer je ?$G_{0}$? ireducibilna podgrupa grupe ?$GL(V)$?, ki je maksimalna za relacijo inkluzije med podgrupami, intranzitivnimi na množici neničelnih vektorjev. Zanima nas razred ?$\Gamma$? vseh grafov premera dve, ki dopuščajo takšno grupo ?$G$? kot ločno-tranzitivno, vozliščno-kvaziprimitivno podgrupo avtomorfizmov. Še posebej obravnavamo tiste grafe, za katere je ?$G_{0}$? podgrupa bodisi ?$\Gamma \mathrm{L}(n,q)$? ali ?$\Gamma \mathrm{Sp}(n, q)$? in je maksimalna v enem od Aschbacherjevih razredov ?$\mathcal{C}_{i}$?, kjer je ?$i \in \{2, 4, 5, 6, 7, 8\}$?. Uspelo nam je določiti vse grafe ?$\Gamma$? ki nastanejo iz ?$G_{0}\leq \Gamma \mathrm{L}(n,q)$?, kjer je ?$i \in \{2, 4, 8\}$?, in ?$G_{0} \leq \Gamma \mathrm{Sp}(n, q)$? kjer je ?$i \in \{2, 8\}$?. Za preostale razrede navedemo potrebne pogoje za to, da ima ?$\Gamma$? premer dve, in v nekaj posebnih podprimerih določimo vse ?$G$?-simetrične grafe premera dveTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije