In this paper, we study finite primitive permutation groups with a small suborbit. Based on the classification result of Quirin [Math. Z. 122 (1971) 267] and Wang [Comm. Algebra 20 (1992) 889], we first produce a precise list of primitive permutation groups with a suborbit of length 4. In particular, we show that there exist no examples of such groups with the point stabiliser of order 2 4 3 6 , clarifying an uncertain question (since 1970s). Then we analyse the orbital graphs of primitive permutation groups with a suborbit of length 3 or of length 4. We obtain a complete classification of vertex-primitive arc-transitive graphs of valency 3 and valency 4, and we prove that there exist no vertex-primitive half-arc-transitive graphs of valency less than 10. Finally, we construct vertexprimitive half-arc-transitive graphs of valency 2k for infinitely many integers k, with 14 as the smallest valency.

A group G is called a CI-group if, for any subsets S, T ⊂ G, whenever two Cayley graphs Cay(G, S) and Cay(G, T ) are isomorphic, there exists an element σ ∈ Aut(G) such that S σ = T . The problem of seeking finite CI-groups is a longstanding open problem in the area of Cayley graphs. This paper contributes towards a complete classification of finite CI-groups. First it is shown that the Frobenius groups of order 4 p and 6 p, and the metacyclic groups of order 9 p of which the centre has order 3 are not CI-groups, where p is an odd prime. Then a shorter explicit list is given of candidates for finite CI-groups. Finally, some new families of finite CI-groups are found, that is, the metacyclic groups of order 4 p (with centre of order 2) and of order 8 p (with centre of order 4) are CI-groups, and a proof is given for the Frobenius group of order 3 p to be a CI-group, where p is a prime.

A characterisation is given of edge-transitive Cayley graphs of valency 4 on odd number of vertices. The characterisation is then applied to solve several problems in the area of edge-transitive graphs: answering a question proposed by Xu [Automorphism groups and isomorphisms of Cayley graphs, Discrete Math. 182 (1998) 309-319] regarding normal Cayley graphs; providing a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex-stabiliser; constructing and characterising a new family of half-transitive graphs. Also this study leads to a construction of the first family of arc-transitive graphs of valency 4 which are non-Cayley graphs and have a 'nice' isomorphic 2-factorisation.

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