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A note on pentavalent \(s\)-transitive graphs. (English) Zbl 1246.05105

Summary: A graph, with a group \(G\) of its automorphisms, is said to be \((G,s)\)-transitive if \(G\) is transitive on \(s\)-arcs but not on \((s+1)\)-arcs of the graph. Let \(X\) be a connected \((G,s)\)-transitive graph for some \(s \geq 1\), and let \(G_{v}\) be the stabilizer of a vertex \(v \in V(X)\) in \(G\). In this paper, we determine the structure of \(G_{v}\) when X has valency 5 and \(G_{v}\) is non-solvable. Together with the results of J.-X. Zhou and Y.-Q. Feng [“On symmetric graphs of valency five”, Discrete Math. 310, No. 12, 1725–1732 (2010; Zbl 1225.05131)], the structure of \(G_{v}\) is completely determined when \(X\) has valency 5. For valency 3 or 4, the structure of \(G_{v}\) is known.

MSC:

05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Citations:

Zbl 1225.05131
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References:

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