On the orders of arc-transitive graphs

M.D.E. Conder, Cai-Heng Li, P. Potočnik

    Research output: Contribution to journalArticle

    5 Citations (Scopus)
    13 Downloads (Pure)

    Abstract

    © 2014 Elsevier Inc. A graph is called arc-transitive (or symmetric) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer k, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is kp for some prime p, and that if d≥4, then there exist only finitely many connected d-valent 2-arc-transitive graphs whose order is kp or kp2 for some prime p. We also prove that there are infinitely many (even) values of k for which there are only finitely many connected 3-valent symmetric graphs of order kp where p is prime.
    Original languageEnglish
    Pages (from-to)167-186
    Number of pages20
    JournalJournal of Algebra
    Volume421
    DOIs
    Publication statusPublished - 1 Jan 2015

    Fingerprint

    Arc-transitive Graph
    Automorphism Group
    Arc of a curve
    Orbit
    Symmetric Graph
    Ordered pair
    Graph in graph theory
    Adjacent
    Path
    Integer

    Cite this

    Conder, M.D.E. ; Li, Cai-Heng ; Potočnik, P. / On the orders of arc-transitive graphs. In: Journal of Algebra. 2015 ; Vol. 421. pp. 167-186.
    @article{8f2ad443632a44a4b31ff4f3afa7a37d,
    title = "On the orders of arc-transitive graphs",
    abstract = "{\circledC} 2014 Elsevier Inc. A graph is called arc-transitive (or symmetric) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer k, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is kp for some prime p, and that if d≥4, then there exist only finitely many connected d-valent 2-arc-transitive graphs whose order is kp or kp2 for some prime p. We also prove that there are infinitely many (even) values of k for which there are only finitely many connected 3-valent symmetric graphs of order kp where p is prime.",
    author = "M.D.E. Conder and Cai-Heng Li and P. Potočnik",
    year = "2015",
    month = "1",
    day = "1",
    doi = "10.1016/j.jalgebra.2014.08.025",
    language = "English",
    volume = "421",
    pages = "167--186",
    journal = "Journal of Algebra",
    issn = "0021-8693",
    publisher = "Academic Press",

    }

    On the orders of arc-transitive graphs. / Conder, M.D.E.; Li, Cai-Heng; Potočnik, P.

    In: Journal of Algebra, Vol. 421, 01.01.2015, p. 167-186.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - On the orders of arc-transitive graphs

    AU - Conder, M.D.E.

    AU - Li, Cai-Heng

    AU - Potočnik, P.

    PY - 2015/1/1

    Y1 - 2015/1/1

    N2 - © 2014 Elsevier Inc. A graph is called arc-transitive (or symmetric) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer k, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is kp for some prime p, and that if d≥4, then there exist only finitely many connected d-valent 2-arc-transitive graphs whose order is kp or kp2 for some prime p. We also prove that there are infinitely many (even) values of k for which there are only finitely many connected 3-valent symmetric graphs of order kp where p is prime.

    AB - © 2014 Elsevier Inc. A graph is called arc-transitive (or symmetric) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer k, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is kp for some prime p, and that if d≥4, then there exist only finitely many connected d-valent 2-arc-transitive graphs whose order is kp or kp2 for some prime p. We also prove that there are infinitely many (even) values of k for which there are only finitely many connected 3-valent symmetric graphs of order kp where p is prime.

    U2 - 10.1016/j.jalgebra.2014.08.025

    DO - 10.1016/j.jalgebra.2014.08.025

    M3 - Article

    VL - 421

    SP - 167

    EP - 186

    JO - Journal of Algebra

    JF - Journal of Algebra

    SN - 0021-8693

    ER -