20.500.12556/RUL-154511 On orders of automorphisms of vertex-transitive graphs In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $\Gamma$, $\Gamma \not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\langle g \rangle$ of length equal to the order of $g$. Moreover, we prove that in this case either $\Gamma$ belongs to a well understood family of exceptional graphs or at least $5/12$ of the vertices of $\Gamma$ belong to a regular orbit of $g$. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms $C$ of a connected $3$-valent vertex-transitive graph $\Gamma$ in terms of the number of vertices of $\Gamma$ and the length of a longest orbit of $C$. graphs automorphism groups vertex-transitive regular orbit cubic tetravalent true false true Angleški jezik Ni določen Članek v reviji 2024-02-19 13:17:19 2024-02-19 13:17:21 2024-05-25 04:13:38 0000-00-00 00:00:00 2024 0 0 Str. 123-153 Vol. 166 May 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 0095-8956 10.1016/j.jctb.2024.01.001 182607619 1-s2.0-S0095895624000029-main.pdf 1-s2.0-S0095895624000029-main.pdf 1 AF102496FFECCFD84678D95EBF0818A6 ae969ba8a0f4d55003777a6b0803f0d52a6a7b00fcd6339a1824c086a5247f8d 81d6a9fe-19bc-11ef-a033-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=183878 https://www.sciencedirect.com/science/article/pii/S0095895624000029 1 d656e222-cf20-11ee-9559-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=180981 Fakulteta za matematiko in fiziko 0 0 0