On the orders of arc-transitive graphs

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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.