On isoperimetric connectivity in vertex-transitive graphs

We shall define the k-isoperimetric connectivity lambda(k) of a regular graph Gamma as the minimum number of arcs originating in a set with cardinality not exceeding half the order of the graph and containing at least k vertices. Clearly lambda(k) less than or equal to dk - e(k), where d is the degree of Gamma and e(k) is the maximal number of edges induced on a set of k vertices. We shall show that Cayley graphs with a prime order and arc-transitive graphs have lambda(k) = dk - e(k), provided that d greater than or equal to 3k - 3. We describe all vertex-transitive graphs where lambda(2) less than or equal to 2d - 3.