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 s\geq 5, and Gv the vertex stabilizer of a vertex v \in V (X). Suppose that Gv is solvable. Weiss (1974) proved that |Gv | p(p−1)^{2}. In this paper, we prove that Gv\cong (\mathbb{Z}_{p}\rtimes \mathbb{Z}_{m})× \mathbb{Z}_{n} for some positive integers m and n such that n | m and m | p − 1., Song-Tao Guo, Hailong Hou, Yong Xu., and Obsahuje seznam literatury
Let $X$ be a finite simple undirected graph with a subgroup $G$ of the full automorphism group ${\rm Aut}(X)$. Then $X$ is said to be $(G,s)$-transitive for a positive integer $s$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs, and $s$-transitive if it is $({\rm Aut}(X),s)$-transitive. Let $G_v$ be a stabilizer of a vertex $v\in V(X)$ in $G$. Up to now, the structures of vertex stabilizers $G_v$ of cubic, tetravalent or pentavalent $(G,s)$-transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers $G_v$ of connected hexavalent $(G,s)$-transitive graphs.