A construction of imprimitive symmetric graphs which are not multicovers of their quotients

Let Σ be a finite X-symmetric graph of valency , and s≥1 an integer. In this article we give a sufficient and necessary condition for the existence of a class of finite imprimitive (X,s)-arc-transitive graphs which have a quotient isomorphic to Σ and are not multicovers of that quotient, together with a combinatorial method, called the double-star graph construction, for constructing such graphs. Moreover, for any X-symmetric graph Γ admitting a nontrivial X-invariant partition B such that Γ is not a multicover of Γ_B, we show that there exists a sequence of -invariant partitions B=B₀,B₁,…,B_m of V(Γ), where m≥1 is an integer, such that B_i is a proper refinement of B_i−1, Γ_Bi is not a multicover of Γ_Bi−1 and Γ_Bi can be reconstructed from Γ_Bi−1 by the double-star graph construction, for i=1,2,…,m, and that either Γ≅Γ_Bm or Γ is a multicover of Γ_Bm.