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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.)

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)

A characterization is given of a class of edge-transitive Cayley graphs, providing methods for constructing Cayley graphs with certain symmetry properties. Various new half-arc transitive graphs are constructed.