Geodesic
1/2-transitive graphs of order $3p$
Journal of Algebraic Combinatorics, Tome 3 (1994) no. 4, pp. 347-355.

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Summary: A graph $X$ is called $vertex-transitive, edge-transitive$, or $arc-transitive$, if the automorphism group of $X$ acts transitively on the set of vertices, edges, or arcs of $X$, respectively. $X$ is said to be $1/2-transitive$, if it is vertex-transitive, edge-transitive, but not arc-transitive. In this paper we determine all 1/2-transitive graphs with $3 p$ vertices, where $p$ is an odd prime. (See Theorem 3.4.)
Keywords: 1/2-transitive graph, metacirculant, factor graph 
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     title = {1/2-transitive graphs of order $3p$},
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Alspach, Brian; Xu, Ming-Yao. 1/2-transitive graphs of order $3p$. Journal of Algebraic Combinatorics, Tome 3 (1994) no. 4, pp. 347-355. http://geodesic.mathdoc.fr/item/JAC_1994__3_4_a2/