Previous |  Up |  Next

Article

Keywords:
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 $m\mid p-1$.
References:
[1] Conder, M., Dobcsányi, P.: Trivalent symmetric graphs on up to 768 vertices. J. Comb. Math. Comb. Comput. 40 (2002), 41-63. MR 1887966 | Zbl 0996.05069
[2] Dixon, J. D., Mortimer, B.: Permutation Groups. Graduate Texts in Mathematics 163 Springer, New York (1996). MR 1409812 | Zbl 0951.20001
[3] Djoković, D. Ž.: A class of finite group-amalgams. Proc. Am. Math. Soc. 80 (1980), 22-26. DOI 10.2307/2042139 | MR 0574502 | Zbl 0441.20015
[4] Djoković, D. Ž., Miller, G. L.: Regular groups of automorphisms of cubic graphs. J. Comb. Theory, Ser. B 29 (1980), 195-230. DOI 10.1016/0095-8956(80)90081-7 | MR 0586434 | Zbl 0385.05040
[5] Feng, Y.-Q., Kwak, J. H.: Cubic symmetric graphs of order a small number times a prime or a prime square. J. Comb. Theory, Ser. B 97 (2007), 627-646. DOI 10.1016/j.jctb.2006.11.001 | MR 2325802 | Zbl 1118.05043
[6] Guo, S.-T., Feng, Y.-Q.: A note on pentavalent {$s$}-transitive graphs. Discrete Math. 312 (2012), 2214-2216. DOI 10.1016/j.disc.2012.04.015 | MR 2926093 | Zbl 1246.05105
[7] Huppert, B.: Endliche Gruppen I. Die Grundlehren der Mathematischen Wissenschaften 134 Springer, Berlin German (1967). DOI 10.1007/978-3-642-64981-3 | MR 0224703 | Zbl 0217.07201
[8] 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 10.1016/j.ejc.2008.10.001 | MR 2514656 | Zbl 1208.05056
[9] Weiss, R.: Presentations for {$(G,s)$}-transitive graphs of small valency. Math. Proc. Camb. Philos. Soc. 101 (1987), 7-20. DOI 10.1017/S0305004100066378 | MR 0877697
[10] Weiss, R.: {$s$}-transitive graphs. Colloq. Math. Soc. János Bolyai 25 North-Holland, Amsterdam (1981), 827-847. Algebraic Methods in Graph Theory, Vol. II L. Lovász et al.; Conf. Szeged, 1978 MR 0642075 | Zbl 0475.05040
[11] Weiss, R.: An application of {$p$}-factorization methods to symmetric graphs. Math. Proc. Camb. Philos. Soc. 85 (1979), 43-48. DOI 10.1017/S030500410005547X | MR 0510398 | Zbl 0392.20002
[12] Weiss, R.: Groups with a {$(B,N)$}-pair and locally transitive graphs. Nagoya Math. J. 74 (1979), 1-21. MR 0535958 | Zbl 0381.20004
[13] Weiss, R. M.: Über symmetrische Graphen, deren Valenz eine Primzahl ist. Math. Z. 136 German (1974), 277-278. MR 0360348 | Zbl 0268.05110
[14] Wielandt, H.: Finite Permutation Groups. Academic Press New York (1964). MR 0183775 | Zbl 0138.02501
[15] Zhou, J.-X., Feng, Y.-Q.: On symmetric graphs of valency five. Discrete Math. 310 (2010), 1725-1732. DOI 10.1016/j.disc.2009.11.019 | MR 2610275 | Zbl 1225.05131
Partner of
EuDML logo