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

Let Σ be a finite X -symmetric graph of valency b ̃ ≥ 2 , 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 m + 1 X -invariant partitions B = B 0 , B 1 , … , B m of V ( Γ ) , where m ≥ 1 is an integer, such that B i is a proper refinement of B i − 1 , Γ B i is not a multicover of Γ B i − 1 and Γ B i can be reconstructed from Γ B i − 1 by the double-star graph construction, for i = 1 , 2 , … , m , and that either Γ ≅ Γ B m or Γ is a multicover of Γ B m .