Hexavalent ( G , s ) -transitive graphs

Song-Tao Guo; Xiao-Hui Hua; Yan-Tao Li

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 4, page 923-931
  • ISSN: 0011-4642

Abstract

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Let X be a finite simple undirected graph with a subgroup G of the full automorphism group 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 ( Aut ( X ) , s ) -transitive. Let G v be a stabilizer of a vertex v ∈ 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.

How to cite

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Guo, Song-Tao, Hua, Xiao-Hui, and Li, Yan-Tao. "Hexavalent $(G,s)$-transitive graphs." Czechoslovak Mathematical Journal 63.4 (2013): 923-931. <http://eudml.org/doc/260795>.

@article{Guo2013,
abstract = {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.},
author = {Guo, Song-Tao, Hua, Xiao-Hui, Li, Yan-Tao},
journal = {Czechoslovak Mathematical Journal},
keywords = {symmetric graph; $s$-transitive graph; $(G,s)$-transitive graph; -arc-transitive graphs; vertex stabilizers},
language = {eng},
number = {4},
pages = {923-931},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Hexavalent $(G,s)$-transitive graphs},
url = {http://eudml.org/doc/260795},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Guo, Song-Tao
AU - Hua, Xiao-Hui
AU - Li, Yan-Tao
TI - Hexavalent $(G,s)$-transitive graphs
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 923
EP - 931
AB - 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.
LA - eng
KW - symmetric graph; $s$-transitive graph; $(G,s)$-transitive graph; -arc-transitive graphs; vertex stabilizers
UR - http://eudml.org/doc/260795
ER -

References

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  1. Bosma, W., Cannon, C., Playoust, C., 10.1006/jsco.1996.0125, J. Symb. Comput. 24 (1997), 235-265. (1997) Zbl0898.68039MR1484478DOI10.1006/jsco.1996.0125
  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). (1985) Zbl0568.20001MR0827219
  3. Dixon, J. D., Mortimer, B., Permutation Groups, Graduate Texts in Mathematics 163 Springer, New York (1996). (1996) Zbl0951.20001MR1409812
  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. Gardiner, A., 10.1093/qmath/24.1.399, Q. J. Math., Oxf. II. Ser. 24 (1973), 399-407. (1973) Zbl0262.05112MR0323617DOI10.1093/qmath/24.1.399
  6. Gardiner, A., 10.1093/qmath/25.1.163, Q. J. Math., Oxf. II. Ser. 25 (1974), 163-167. (1974) Zbl0305.05111MR0412015DOI10.1093/qmath/25.1.163
  7. Gardiner, A., 10.1093/qmath/27.3.313, Q. J. Math., Oxf. II. Ser. 27 (1976), 313-323. (1976) Zbl0337.05117MR0498228DOI10.1093/qmath/27.3.313
  8. 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
  9. Li, C. H., 10.1090/S0002-9947-01-02768-4, Trans. Am. Math. Soc. (electronic) 353 (2001), 3511-3529. (2001) MR1837245DOI10.1090/S0002-9947-01-02768-4
  10. 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
  11. Potočnik, P., Spiga, P., Verret, G., 10.1017/S0004972710002078, Bull. Aust. Math. Soc. 84 (2011), 79-89. (2011) Zbl1222.05102MR2817661DOI10.1017/S0004972710002078
  12. Stroth, G., Weiss, R., A new construction of the group R u , Q. J. Math., Oxf. II. Ser. 41 (1990), 237-243. (1990) Zbl0695.20015MR1053664
  13. Tutte, W. T., 10.1017/S0305004100023720, Proc. Camb. Philos. Soc. 43 (1947), 459-474. (1947) Zbl0029.42401MR0021678DOI10.1017/S0305004100023720
  14. Weiss, R. M., 10.1016/0095-8956(74)90049-5, J. Comb. Theory, Ser. B 17 (1974), 59-64 German. (1974) Zbl0298.05130MR0369151DOI10.1016/0095-8956(74)90049-5
  15. Weiss, R. M., 10.1007/BF01214131, Math. Z. 136 (1974), 277-278 German. (1974) Zbl0268.05110MR0360348DOI10.1007/BF01214131
  16. Weiss, R. M., 10.1017/S030500410005547X, Math. Proc. Camb. Philos. Soc. 85 (1979), 43-48. (1979) Zbl0392.20002MR0510398DOI10.1017/S030500410005547X
  17. Weiss, R. M., 10.1007/BF02579337, Combinatorica 1 (1981), 309-311. (1981) Zbl0486.05032MR0637836DOI10.1007/BF02579337
  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. (1981) Zbl0475.05040MR0642075
  19. Weiss, R. M., 10.1017/S0305004100066378, Math. Proc. Camb. Phil. Soc. 101 (1987), 7-20. (1987) MR0877697DOI10.1017/S0305004100066378
  20. Wielandt, H., Finite Permutation Groups. Translated from the German by R. Bercov, Academic Press, New York (1964). (1964) MR0183775
  21. 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

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