On Half-Transitive Metacirculant Graphs of Prime-Power Order

In this paper, infinitely many graphs of large valency which are half-transitive (that is vertex- and edge-transitive but not arc-transitive) are constructed, and a complete classification is given of half-transitive metacirculant graphs of order a p-power and valency less than 2p, where p is a prime. In particular, it is shown that, for any odd prime p, integers n⩾3 and k⩾2 such that k divides p−1, there are exactly (pn−2−1)/2+pn−3−1 nonisomorphic connected half-transitive metacirculants of order pn and valency 2k.