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Discussiones Mathematicae Graph Theory 26(3) (2006) 359-368
DOI: https://doi.org/10.7151/dmgt.1328

ARC-TRANSITIVE AND s-REGULAR CAYLEY GRAPHS OF VALENCY FIVE ON ABELIAN GROUPS

Mehdi Alaeiyan

Department of Mathematics
Iran University of Science and Technology
Narmak, Tehran 16844, Iran
e-mail: alaeiyan@iust.ac.ir

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.

Keywords: Cayley graph, normal Cayley graph, arc-transitive, s-regular Cayley graph.

2000 Mathematics Subject Classification: 05C25, 20B25.

References

[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.
[2]	N. Biggs, Algebraic Graph Theory (Cambridge University Press, 1974).
[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.
[4]	J.L. Berggren, An algebraic characterization of symmetric graph with p point, Bull. Aus. Math. Soc. 158 (1971) 247-256.
[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.
[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.
[7]	J.D. Dixon and B. Mortimer, Permutation Groups (Springer-Verlag, 1996).
[8]	C.D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981) 243-256, doi: 10.1007/BF02579330.
[9]	C.D. Godsil and G. Royle, Algebric Graph Theory (Springer-Verlag, 2001).
[10]	H. Wielandt, Finite Permutation Group (Academic Press, New York, 1964).
[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.
[12]	Ming-Yao Xu, A note on one-regular graphs of valency four, Chinese Science Bull. 45 (2000) 2160-2162.
[13]	Ming-Yao Xu, Hyo-Seob Sim and Youg-Gheel Baik, Arc-transitive circulant digraphs of odd prime-power order, (summitted).
[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.

Received 29 November 2005
Revised 5 June 2006