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Keywords:
symmetric graph; $s$-transitive graph; $(G,s)$-transitive graph
symmetric graph; $s$-transitive graph; $(G,s)$-transitive graph
Summary:
A graph $X$, with a group $G$ of automorphisms of $X$, is said to be $(G,s)$-transitive, for some $s\geq 1$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs. Let $X$ be a connected $(G,s)$-transitive graph of prime valency $p\geq 5$, and $G_v$ the vertex stabilizer of a vertex $v\in V(X)$. Suppose that $G_v$ is solvable. Weiss (1974) proved that $|G_v|\mid p(p-1)^2$. In this paper, we prove that $G_v\cong (\mathbb Z_p\rtimes \mathbb Z_m)\times \mathbb Z_n$ for some positive integers $m$ and $n$ such that $n\div m$ and $m\mid p-1$.
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