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

top
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

top

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

top
  1. [1] B. Alspach, M. Conder, D. Marusic and Ming-Yao Xu, A classification of 2-arc-transitive circulant, J. Algebraic Combin. 5 (1996) 83-86, doi: 10.1023/A:1022456615990. Zbl0849.05034
  2. [2] N. Biggs, Algebraic Graph Theory (Cambridge University Press, 1974). Zbl0284.05101
  3. [3] Y.G. Baik, Y.Q. Feng and H.S. Sim, The normality of Cayley graphs of finite Abelian groups with valency 5, System Science and Mathematical Science 13 (2000) 420-431. Zbl0973.05038
  4. [4] J.L. Berggren, An algebraic characterization of symmetric graph with p point, Bull. Aus. Math. Soc. 158 (1971) 247-256. 
  5. [5] C.Y. Chao, On the classification of symmetric graph with a prime number of vertices, Trans. Amer. Math. Soc. 158 (1971) 247-256, doi: 10.1090/S0002-9947-1971-0279000-7. Zbl0217.02403
  6. [6] C.Y. Chao and J. G. Wells, A class of vertex-transitive digraphs, J. Combin. Theory (B) 14 (1973) 246-255, doi: 10.1016/0095-8956(73)90007-5. Zbl0245.05109
  7. [7] J.D. Dixon and B. Mortimer, Permutation Groups (Springer-Verlag, 1996). Zbl0951.20001
  8. [8] C.D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981) 243-256, doi: 10.1007/BF02579330. Zbl0489.05028
  9. [9] C.D. Godsil and G. Royle, Algebric Graph Theory (Springer-Verlag, 2001). 
  10. [10] H. Wielandt, Finite Permutation Group (Academic Press, New York, 1964). Zbl0138.02501
  11. [11] Ming-Yao Xu and Jing Xu, Arc-transitive Cayley graph of valency at most four on Abelian Groups, Southest Asian Bull. Math. 25 (2001) 355-363, doi: 10.1007/s10012-001-0355-z. Zbl0993.05086
  12. [12] Ming-Yao Xu, A note on one-regular graphs of valency four, Chinese Science Bull. 45 (2000) 2160-2162. 
  13. [13] Ming-Yao Xu, Hyo-Seob Sim and Youg-Gheel Baik, Arc-transitive circulant digraphs of odd prime-power order, (summitted). Zbl1050.05064
  14. [14] Ming-Yao Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998) 309-319, doi: 10.1016/S0012-365X(97)00152-0. Zbl0887.05025

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.