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A. DELGADO AND R. WEISS 1. Introduction Let F be a connected, undirected graph and let G be a subgroup of aut (F) such that \G \ < oo for each vertex a. For each vertex a, let F(a) denote the set of vertices of F adjacent to a. An s-path in F is a sequence (a , a ... , a ) of s+ 1 vertices a. such 0 1? s t that a,er(a^_ ) for 1 ^ i ^ s and a, ^ a,_ for 2 < i ^ s. Suppose that for some 1 2 O 0, the graph F is s-transitive with respect to G, that is, G acts transitively on the set of 5-paths in F but intransitively on the set of (s+ l)-paths. Let k denote the valency of F, that is, k = |F(a)|, a an arbitrary vertex of F, and suppose that k ^ 3. By [10], s ^ 5 or s = 7 and, if s ^ 4, then k = q+\, where ^ is a power of some prime p, with p = \(s—\) for 5 = 5,7. (*) Let g denote the girth of F, that is, the
Bulletin of the London Mathematical Society – Wiley
Published: May 1, 1989
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