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

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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 ≅ ( ℤ p ⋊ ℤ m ) × ℤ n for some positive integers m and n such that n div m and m ∣ p - 1 .

How to cite

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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

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