Arc-transitive and s-regular Cayley graphs of valency five on Abelian groups

Mehdi Alaeiyan

Discussiones Mathematicae Graph Theory (2006)

  • Volume: 26, Issue: 3, page 359-368
  • ISSN: 2083-5892

Abstract

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Let G be a finite group, and let 1 G ∉ S ⊆ G . A Cayley di-graph Γ = Cay(G,S) of G relative to S is a di-graph with a vertex set G such that, for x,y ∈ G, the pair (x,y) is an arc if and only if y x - 1 ∈ S . Further, if S = S - 1 : = s - 1 | s ∈ S , then Γ is undirected. Γ is conected if and only if G = ⟨s⟩. A Cayley (di)graph Γ = Cay(G,S) is called normal if the right regular representation of G is a normal subgroup of the automorphism group of Γ. A graph Γ is said to be arc-transitive, if Aut(Γ) is transitive on an arc set. Also, a graph Γ is s-regular if Aut(Γ) acts regularly on the set of s-arcs. In this paper, we first give a complete classification for arc-transitive Cayley graphs of valency five on finite Abelian groups. Moreover, we classify s-regular Cayley graph with valency five on an abelian group for each s ≥ 1.

How to cite

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Mehdi Alaeiyan. "Arc-transitive and s-regular Cayley graphs of valency five on Abelian groups." Discussiones Mathematicae Graph Theory 26.3 (2006): 359-368. <http://eudml.org/doc/270263>.

@article{MehdiAlaeiyan2006,
abstract = {Let G be a finite group, and let $1_G ∉ S ⊆ G$. A Cayley di-graph Γ = Cay(G,S) of G relative to S is a di-graph with a vertex set G such that, for x,y ∈ G, the pair (x,y) is an arc if and only if $yx^\{-1\} ∈ S$. Further, if $S = S^\{-1\}:= \{s^\{-1\}|s ∈ S\}$, then Γ is undirected. Γ is conected if and only if G = ⟨s⟩. A Cayley (di)graph Γ = Cay(G,S) is called normal if the right regular representation of G is a normal subgroup of the automorphism group of Γ. A graph Γ is said to be arc-transitive, if Aut(Γ) is transitive on an arc set. Also, a graph Γ is s-regular if Aut(Γ) acts regularly on the set of s-arcs. In this paper, we first give a complete classification for arc-transitive Cayley graphs of valency five on finite Abelian groups. Moreover, we classify s-regular Cayley graph with valency five on an abelian group for each s ≥ 1.},
author = {Mehdi Alaeiyan},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Cayley graph; normal Cayley graph; arc-transitive; s-regular Cayley graph},
language = {eng},
number = {3},
pages = {359-368},
title = {Arc-transitive and s-regular Cayley graphs of valency five on Abelian groups},
url = {http://eudml.org/doc/270263},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Mehdi Alaeiyan
TI - Arc-transitive and s-regular Cayley graphs of valency five on Abelian groups
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 3
SP - 359
EP - 368
AB - Let G be a finite group, and let $1_G ∉ S ⊆ G$. A Cayley di-graph Γ = Cay(G,S) of G relative to S is a di-graph with a vertex set G such that, for x,y ∈ G, the pair (x,y) is an arc if and only if $yx^{-1} ∈ S$. Further, if $S = S^{-1}:= {s^{-1}|s ∈ S}$, then Γ is undirected. Γ is conected if and only if G = ⟨s⟩. A Cayley (di)graph Γ = Cay(G,S) is called normal if the right regular representation of G is a normal subgroup of the automorphism group of Γ. A graph Γ is said to be arc-transitive, if Aut(Γ) is transitive on an arc set. Also, a graph Γ is s-regular if Aut(Γ) acts regularly on the set of s-arcs. In this paper, we first give a complete classification for arc-transitive Cayley graphs of valency five on finite Abelian groups. Moreover, we classify s-regular Cayley graph with valency five on an abelian group for each s ≥ 1.
LA - eng
KW - Cayley graph; normal Cayley graph; arc-transitive; s-regular Cayley graph
UR - http://eudml.org/doc/270263
ER -

References

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