A class of symmetric graphs with 2-arc transitive quotients

Let Γ be a finite X-symmetric graph with a nontrivial Xinvariant partition B on V (Γ) such that ΓB is a connected (X, 2)-arctransitive graph and Γ is not a multicover of ΓB. A characterization of (Γ,X,B) was given in [20] for the case where |Γ(C) ∩ B| = 2 for B ∈ B and C ∈ ΓB(B). This motivates us to investigate the case where |Γ(C) ∩ B| = 3, that is, Γ[B,C] is isomorphic to one of 3K2, K3,3 − 3K2 and K3,3. This investigation requires a study on (X, 2)arc-transitive graphs of valency 4 or 7. Based on the results in [14], we give a characterization of tetravalent (X, 2)-arc-transitive graphs; and as a byproduct, we prove that every tetravalent (X, 2)-transitive graph is either the complete graph on 5 vertices or a near n-gonal graph for some n ≥ 4. We show that a heptavalent (X, 2)-arc-transitive graph Σ can occur as ΓB if and only if X Σ(τ) τ ∼= PSL(3, 2) for τ ∈ V (Σ).