Abstract
It is proved that if p is a prime, k and m⩽ p are positive integers, and I is a vertex symmetric diagraph of order p k or mp, then Γ has an automorphism all of whose orbits have cardinality p. Vertex symmetric graphs of order 2 p such that 2 p−1 is not the square of a composite integer and vertex symmetric digraphs of order p k are characterised.
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