Classification of half-arc-transitive graphs of order

A vertex-transitive graph X is said to be half-arc-transitive if its automorphism group acts transitively on the set of edges of X but does not act transitively on the set of arcs of X . A classification of half-arc-transitive graphs on 4 p vertices, where p is a prime, is given. Apart from an obvious infinite family of metacirculants, which exist for p ≡ 1 ( mod 4 ) and have been known before, there is an additional somewhat unique family of half-arc-transitive graphs of order 4 p and valency 12 ; the latter exists only when p ≡ 1 ( mod 6 ) is of the form 2 2 k + 2 k + 1 , k > 1 .