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Keywords:
arc-transitive graph; symmetric graph; $s$-regular graph
Summary:
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.
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