# On the orders of arc-transitive graphs

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

Research output: Contribution to journalArticle

5 Citations (Scopus)

### 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 language English 167-186 20 Journal of Algebra 421 https://doi.org/10.1016/j.jalgebra.2014.08.025 Published - 1 Jan 2015

### Fingerprint

Arc-transitive Graph
Automorphism Group
Arc of a curve
Orbit
Symmetric Graph
Ordered pair
Graph in graph theory
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.
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",

}

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 -