Symmetric graphs of valency seven and their basic normal quotient graphs

Abstract

A graph Γ is basic if AutΓ has no normal subgroup N = 1 such that Γ is a normal cover of the normal quotient graph Γ 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 ≡ 1(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 = p < q primes, partially generalizing Theorem 1 of Conder, Li and Potočnik [On the orders of arctransitive graphs, J. Algebra 421 (2015), 167-186].