20.500.12556/RUP-17626 Vertex-transitive graphs and their arc-types Vozliščno tranzitivni grafi in njihovi ločni tipi Let ▫$X$▫ be a finite vertex-transitive graph of valency ▫$d$▫, and let ▫$A$▫ be the full automorphism group of ▫$X$▫. Then the arc-type of ▫$X$▫ is defined in terms of the sizes of the orbits of the stabiliser ▫$A_v$▫ of a given vertex ▫$v$▫ on the set of arcs incident with ▫$v$▫. Such an orbit is said to be self-paired if it is contained in an orbit ▫$\Delta$▫ of ▫$A$▫ on the set of all arcs of v$X$▫ such that v$\Delta$▫ is closed under arc-reversal. The arc-type of ▫$X$▫ is then the partition of ▫$d$▫ as the sum ▫$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$▫, where ▫$n_1, n_2, \dots, n_t$▫ are the sizes of the self-paired orbits, and ▫$m_1,m_1, m_2,m_2, \dots, m_s,m_s$▫ are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two "relatively prime" graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of ▫$1+1$▫ and ▫$(1+1)$▫, every partition as defined above is \emph{realisable}, in the sense that there exists at least one vertex-transitive graph with the given partition as its arc-type. Naj bo ▫$X$▫ končen vozliščno-tranzitiven graf valence ▫$d$▫, in naj bo ▫$A$▫ polna grupa avtomorfizmov grafa ▫$X$▫. Potem je ločni tip grafa ▫$X$▫ definiran v smislu velikosti orbit stabilizatorja ▫$A_v$▫ danega vozlišča ▫$v$▫ na množici lokov incidentnih z ▫$v$▫. Za takšno orbito pravimo, da je sebi-prirejena, če je vsebovana v orbiti ▫$\Delta$▫ grupe ▫$A$▫ na množici vseh takšnih lokov grafa ▫$X$▫, za katere je orbita ▫$\Delta$▫ zaprta za obračanje lokov. Ločni tip grafa ▫$X$▫ je tedaj razčlenitev števila ▫$d$▫ v vsoto ▫$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$▫, kjer so ▫$n_1, n_2, \dots, n_t$▫ velikosti sebi-prirejenih orbit, ▫$m_1,m_1, m_2,m_2, \dots, m_s,m_s$▫, ms pa so velikosti sebi-neprirejenih orbit, v padajočem redu. V tem članku najdemo ločne tipe več družin grafov. Pokažemo tudi, da je ločni tip kartezičnega produkta dveh "tujih si" grafov naravna vsota njunih ločnih tipov. Potem na podlagi teh opažanj pokažemo, da je, z izjemo ▫$1+1$▫ in ▫$(1+1)$▫, vsaka particija, kot je definirana zgoraj, realizabilna, v tem smislu, da obstaja vsaj en vozliščno-tranzitiven graf z dano razčlenitvijo kot svojim ločnim tipom. symmetry type vertex-transitive graph arc-transitive graph Cayley graph cartesian product covering graph tip simetrije vozliščno tranzitiven graf ločno trazitiven graf Cayleyjev graf kartezični produkt krovni graf true true false Angleški jezik Slovenski jezik Neznano 2022-01-03 01:09:52 2022-01-03 01:09:52 2024-03-01 14:39:22 0000-00-00 00:00:00 2017 0 0 str. 383-413 no. 2 Vol. 12 2017 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17:512.54 1855-3966 18064217 RAZ_Conder_Marston_D._E._i2017.pdf RAZ_Conder_Marston_D._E._i2017.pdf 1 0190B608AC47F66AC3D1800AA68C2419 b7c192901f7556e0bac21b334f35831ba544f217fbc9e52ea01f85a4d6b50928 78269c4a-c3d6-11ed-b5df-005056bf369b https://repozitorij.upr.si/Dokument.php?lang=slv&id=25943 Založba Univerze na Primorskem 0 0 0