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

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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 1G ∉ 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.

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Received 29 November 2005
Revised 5 June 2006