| Title: | Vertex-transitive graphs and their arc-types |
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| Authors: | ID Conder, Marston D. E. (Author)
ID Pisanski, Tomaž (Author)
ID Žitnik, Arjana (Author) |
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| Files: |  RAZ_Conder_Marston_D._E._i2017.pdf (475,17 KB)
MD5: 0190B608AC47F66AC3D1800AA68C2419
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| Language: | English |
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| Work type: | Unknown |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | ZUP - University of Primorska Press
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| Abstract: | 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. |
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| Keywords: | symmetry type, vertex-transitive graph, arc-transitive graph, Cayley graph, cartesian product, covering graph |
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| Year of publishing: | 2017 |
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| Number of pages: | str. 383-413 |
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| Numbering: | Vol. 12, no. 2 |
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| PID: | 20.500.12556/RUP-17626  |
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| UDC: | 519.17:512.54 |
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| ISSN on article: | 1855-3966 |
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| COBISS.SI-ID: | 18064217 |
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| Publication date in RUP: | 03.01.2022 |
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| Views: | 850 |
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| Downloads: | 18 |
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| Metadata: |     |
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