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Let r be a simple undirected graph and G a subgroup of Aut r. r is said to be G-symmetric, if G acts transitively on the set of ordered adjacent pairs of vertices of r; r is said to be symmetric if it is Autr-symmetric. In this paper we give a complete classification for symmetric graphs of order 30. (See Theorem 10.)

A graph is 1-arc-regular if its full automorphism group acts regularly on the set of its arcs. In this paper, we investigate 1-arc-regular graphs of prime valency, especially of valency 3. First, we prove that if G is a soluble group then a (G, 1)-arc-regular graph must be a Cayley graph of a subgroup of G. Next we consider trivalent Cayley graphs of a… (More)

For a positive integer s, a graph Γ is called s-arc transitive if its full automorphism group AutΓ acts transitively on the set of s-arcs of Γ. Given a group G and a subset S of G with S = S −1 and 1 / ∈ S, let Γ = Cay(G, S) be the Cayley graph of G with respect to S and G R the set of right translations of G on G. Then G R forms a regular subgroup of AutΓ.… (More)