Geodesic
Arc-transitive and s-regular Cayley graphs of valency five on Abelian groups
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 3, pp. 359-368 Cet article a éte moissonné depuis la source Library of Science

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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 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 
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Alaeiyan, Mehdi. Arc-transitive and s-regular Cayley graphs of valency five on Abelian groups. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 3, pp. 359-368. http://geodesic.mathdoc.fr/item/DMGT_2006_26_3_a0/

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