Valency seven symmetric graphs of order 2<Emphasis Type="Italic">pq</Emphasis>

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected valency seven symmetric graphs of order 2pq are classified, where p, q are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order 4p, and that for odd primes p and q, there is an infinite family of connected valency seven one-regular graphs of order 2pq with solvable automorphism groups, and there are four sporadic ones with nonsolvable automorphism groups, which is 1, 2, 3-arc transitive, respectively. In particular, one of the four sporadic ones is primitive, and the other two of the four sporadic ones are bi-primitive.