A graph $\Gamma$ is basic if Aut$\Gamma$ has no normal subgroup $N\ne1$ such that $\Gamma$ is a normal cover of the normal quotient graph $\Gamma_N$. In this paper, we completely determine the basic normal quotient graphs of all connected 7-valent symmetric graphs of order $2pq^n$ with $p < q$ odd primes, which consist of an infinite family of dihedrants of order $2p$ with $p\equiv1$(mod 7), and 6 specific graphs with order at most 310. As a consequence, it shows that, for any given positive integer n, there are only finitely many connected 2-arc-transitive 7-valent graphs of order $2pq^n$ with $7\ne p<q$ primes, partially generalizing Theorem 1 of Conder, Li and Potocnik [On the orders of arc-transitive graphs, J. Algebra 421 (2015), 167-186].

Symmetric graphs of valency seven and their basic normal quotient graphs

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Abstract In this paper, we classify symmetric basic prime-valent graphs of order four times an odd prime power. As an application, prime-valent symmetric graphs of order four times a prime or prime square are specifically determined. We remark that special cases for valency 5 and 7 were characterized by Yang etc. (Eur. J. Combin. 80:236–246, 2019) and Pan and Yin (J. Algebr. Appl. 17(5):1850093, 2018).

On prime-valent symmetric graphs of order four times an odd prime power

A graph Γ is called G-basic if G is quasiprimitive or bi-quasiprimitive on the vertex set of Γ, where G⩽Aut(Γ). It is known that locally primitive vertex-transitive graphs are normal covers of basic ones. In this paper, a complete classification of the basic locally primitive vertex-transitive graph of order 2p2 is given, where p is an odd prime.

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The Basic Locally Primitive Graphs of Order Twice a Prime Square

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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Li Chen

Papers

Valency seven symmetric graphs of order 2pq