| Title: | On orders of automorphisms of vertex-transitive graphs |
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| Authors: | ID Potočnik, Primož (Author)
ID Toledo, Micael (Author)
ID Verret, Gabriel (Author) |
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| Files: |  URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0095895624000029
 PDF - Presentation file, download (573,20 KB)
MD5: AC65152D691832B4F4B1A64AC61488AE
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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 $C$. |
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| Keywords: | graphs, automorphism groups, vertex-transitive, regular orbit, cubic, tetravalent |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.05.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | str. 123-153 |
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| Numbering: | Vol. 166 |
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| PID: | 20.500.12556/DiRROS-18208  |
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| UDC: | 519.17 |
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| ISSN on article: | 0095-8956 |
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| DOI: | 10.1016/j.jctb.2024.01.001 |
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| COBISS.SI-ID: | 182607619 |
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| Publication date in DiRROS: | 19.02.2024 |
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| Views: | 1177 |
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| Downloads: | 615 |
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| Metadata: |    |
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