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

Let ΣΣ be a finite XX-symmetric graph of valency b̃≥2, and s≥1s≥1 an integer. In this article we give a sufficient and necessary condition for the existence of a class of finite imprimitive (X,s)(X,s)-arc-transitive graphs which have a quotient isomorphic to ΣΣ 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 XX-symmetric graph ΓΓ admitting a nontrivial XX-invariant partition BB such that ΓΓ is not a multicover of ΓBΓB, we show that there exists a sequence of m+1X-invariant partitions B=B0,B1,…,BmB=B0,B1,…,Bm of V(Γ)V(Γ), where m≥1m≥1 is an integer, such that BiBi is a proper refinement of Bi−1Bi−1, ΓBiΓBi is not a multicover of ΓBi−1ΓBi−1 and ΓBiΓBi can be reconstructed from ΓBi−1ΓBi−1 by the double-star graph construction, for i=1,2,…,mi=1,2,…,m, and that either Γ≅ΓBmΓ≅ΓBm or ΓΓ is a multicover of ΓBmΓBm.