Geodesic
Hexavalent $(G,s)$-transitive graphs
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 923-931
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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.
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.
DOI : 10.1007/s10587-013-0062-9
Classification : 05C25, 20B25
Keywords: symmetric graph; $s$-transitive graph; $(G, s)$-transitive graph 
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Guo, Song-Tao; Hua, Xiao-Hui; Li, Yan-Tao. Hexavalent $(G,s)$-transitive graphs. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 923-931. doi: 10.1007/s10587-013-0062-9

[1] Bosma, W., Cannon, C., Playoust, C.: The Magma algebra system. I: The user language. J. Symb. Comput. 24 (1997), 235-265. | DOI | MR | Zbl

[2] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups. Maximal subgroups and ordinary characters for simple groups Clarendon Press, Oxford (1985). | MR | Zbl

[3] Dixon, J. D., Mortimer, B.: Permutation Groups. Graduate Texts in Mathematics 163 Springer, New York (1996). | MR | Zbl

[4] Djoković, D. Ž., Miller, G. L.: Regular groups of automorphisms of cubic graphs. J. Comb. Theory, Ser. B 29 (1980), 195-230. | DOI | MR | Zbl

[5] Gardiner, A.: Arc transitivity in graphs. Q. J. Math., Oxf. II. Ser. 24 (1973), 399-407. | DOI | MR | Zbl

[6] Gardiner, A.: Arc transitivity in graphs. II. Q. J. Math., Oxf. II. Ser. 25 (1974), 163-167. | DOI | MR | Zbl

[7] Gardiner, A.: Arc transitivity in graphs. III. Q. J. Math., Oxf. II. Ser. 27 (1976), 313-323. | DOI | MR | Zbl

[8] Guo, S. T., Feng, Y. Q.: A note on pentavalent $s$-transitive graphs. Discrete Math. 312 (2012), 2214-2216. | DOI | MR | Zbl

[9] Li, C. H.: The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\geq 4$. Trans. Am. Math. Soc. (electronic) 353 (2001), 3511-3529. | DOI | MR

[10] Potočnik, P.: A list of $4$-valent $2$-arc-transitive graphs and finite faithful amalgams of index $(4,2)$. Eur. J. Comb. 30 (2009), 1323-1336. | DOI | MR | Zbl

[11] Potočnik, P., Spiga, P., Verret, G.: Tetravalent arc-transitive graphs with unbounded vertex-stabilizers. Bull. Aust. Math. Soc. 84 (2011), 79-89. | DOI | MR | Zbl

[12] Stroth, G., Weiss, R.: A new construction of the group $ Ru$. Q. J. Math., Oxf. II. Ser. 41 (1990), 237-243. | MR | Zbl

[13] Tutte, W. T.: A family of cubical graphs. Proc. Camb. Philos. Soc. 43 (1947), 459-474. | DOI | MR | Zbl

[14] Weiss, R. M.: Über symmetrische Graphen vom Grad fünf. J. Comb. Theory, Ser. B 17 (1974), 59-64 German. | DOI | MR | Zbl

[15] Weiss, R. M.: Über symmetrische Graphen, deren Valenz eine Primzahl ist. Math. Z. 136 (1974), 277-278 German. | DOI | MR | Zbl

[16] Weiss, R. M.: An application of $p$-factorization methods to symmetric graphs. Math. Proc. Camb. Philos. Soc. 85 (1979), 43-48. | DOI | MR | Zbl

[17] Weiss, R. M.: The nonexistence of $8$-transitive graphs. Combinatorica 1 (1981), 309-311. | DOI | MR | Zbl

[18] Weiss, R. M.: $s$-transitive graphs. Algebraic Methods in Graph Theory, Vol. I, II Colloq. Math. Soc. Janos Bolyai 25 (Szeged, 1978) (1981), 827-847 North-Holland, Amsterdam. | MR | Zbl

[19] Weiss, R. M.: Presentations for $(G,s)$-transitive graphs of small valency. Math. Proc. Camb. Phil. Soc. 101 (1987), 7-20. | DOI | MR

[20] Wielandt, H.: Finite Permutation Groups. Translated from the German by R. Bercov. Academic Press, New York (1964). | MR

[21] Zhou, J. X., Feng, Y. Q.: On symmetric graphs of valency five. Discrete Math. 310 (2010), 1725-1732. | DOI | MR | Zbl

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