A note on solvable vertex stabilizers of s -transitive graphs of prime valency

Song-Tao Guo; Hailong Hou; Yong Xu

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 3, page 781-785
  • ISSN: 0011-4642

Abstract

top
A graph X , with a group G of automorphisms of X , is said to be ( G , s ) -transitive, for some s 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 5 , and G v the vertex stabilizer of a vertex v V ( X ) . Suppose that G v is solvable. Weiss (1974) proved that | G v | p ( p - 1 ) 2 . In this paper, we prove that G v ( p m ) × n for some positive integers m and n such that n div m and m p - 1 .

How to cite

top

Guo, Song-Tao, Hou, Hailong, and Xu, Yong. "A note on solvable vertex stabilizers of $s$-transitive graphs of prime valency." Czechoslovak Mathematical Journal 65.3 (2015): 781-785. <http://eudml.org/doc/271813>.

@article{Guo2015,
abstract = {A graph $X$, with a group $G$ of automorphisms of $X$, is said to be $(G,s)$-transitive, for some $s\ge 1$, 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 $p\ge 5$, 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 $n\operatorname\{div\}m$ and $m\mid p-1$.},
author = {Guo, Song-Tao, Hou, Hailong, Xu, Yong},
journal = {Czechoslovak Mathematical Journal},
keywords = {symmetric graph; $s$-transitive graph; $(G,s)$-transitive graph},
language = {eng},
number = {3},
pages = {781-785},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on solvable vertex stabilizers of $s$-transitive graphs of prime valency},
url = {http://eudml.org/doc/271813},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Guo, Song-Tao
AU - Hou, Hailong
AU - Xu, Yong
TI - A note on solvable vertex stabilizers of $s$-transitive graphs of prime valency
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 3
SP - 781
EP - 785
AB - A graph $X$, with a group $G$ of automorphisms of $X$, is said to be $(G,s)$-transitive, for some $s\ge 1$, 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 $p\ge 5$, 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 $n\operatorname{div}m$ and $m\mid p-1$.
LA - eng
KW - symmetric graph; $s$-transitive graph; $(G,s)$-transitive graph
UR - http://eudml.org/doc/271813
ER -

References

top
  1. Conder, M., Dobcsányi, P., Trivalent symmetric graphs on up to 768 vertices, J. Comb. Math. Comb. Comput. 40 (2002), 41-63. (2002) Zbl0996.05069MR1887966
  2. Dixon, J. D., Mortimer, B., Permutation Groups, Graduate Texts in Mathematics 163 Springer, New York (1996). (1996) Zbl0951.20001MR1409812
  3. Djoković, D. Ž., 10.2307/2042139, Proc. Am. Math. Soc. 80 (1980), 22-26. (1980) Zbl0441.20015MR0574502DOI10.2307/2042139
  4. Djoković, D. Ž., Miller, G. L., 10.1016/0095-8956(80)90081-7, J. Comb. Theory, Ser. B 29 (1980), 195-230. (1980) Zbl0385.05040MR0586434DOI10.1016/0095-8956(80)90081-7
  5. Feng, Y.-Q., Kwak, J. H., 10.1016/j.jctb.2006.11.001, J. Comb. Theory, Ser. B 97 (2007), 627-646. (2007) Zbl1118.05043MR2325802DOI10.1016/j.jctb.2006.11.001
  6. Guo, S.-T., Feng, Y.-Q., 10.1016/j.disc.2012.04.015, Discrete Math. 312 (2012), 2214-2216. (2012) Zbl1246.05105MR2926093DOI10.1016/j.disc.2012.04.015
  7. Huppert, B., 10.1007/978-3-642-64981-3, Die Grundlehren der Mathematischen Wissenschaften 134 Springer, Berlin German (1967). (1967) Zbl0217.07201MR0224703DOI10.1007/978-3-642-64981-3
  8. Potočnik, P., 10.1016/j.ejc.2008.10.001, Eur. J. Comb. 30 (2009), 1323-1336. (2009) Zbl1208.05056MR2514656DOI10.1016/j.ejc.2008.10.001
  9. Weiss, R., 10.1017/S0305004100066378, Math. Proc. Camb. Philos. Soc. 101 (1987), 7-20. (1987) MR0877697DOI10.1017/S0305004100066378
  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 (1981) Zbl0475.05040MR0642075
  11. Weiss, R., 10.1017/S030500410005547X, Math. Proc. Camb. Philos. Soc. 85 (1979), 43-48. (1979) Zbl0392.20002MR0510398DOI10.1017/S030500410005547X
  12. Weiss, R., Groups with a ( B , N ) -pair and locally transitive graphs, Nagoya Math. J. 74 (1979), 1-21. (1979) Zbl0381.20004MR0535958
  13. Weiss, R. M., Über symmetrische Graphen, deren Valenz eine Primzahl ist, Math. Z. 136 German (1974), 277-278. (1974) Zbl0268.05110MR0360348
  14. Wielandt, H., Finite Permutation Groups, Academic Press New York (1964). (1964) Zbl0138.02501MR0183775
  15. Zhou, J.-X., Feng, Y.-Q., 10.1016/j.disc.2009.11.019, Discrete Math. 310 (2010), 1725-1732. (2010) Zbl1225.05131MR2610275DOI10.1016/j.disc.2009.11.019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.