Let Σ be a finite X-symmetric graph of valencyb ≥ 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 partitionsis 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 Bm or Γ is a multicover of Γ Bm .Let Γ be a finite X-symmetric graph and B a partition ofA finite X-symmetric graph Γ is imprimitive if its vertex set admits a nontrivial X-invariant partition B; otherwise it is called primitive. In the imprimitive case, the quotient graph with respect to B is defined to have vertex set B in which two blocks are adjacent if and only if there exists at least one edge of Γ between them. In this article we always assume that Γ B is nonempty, that is, the valency b of Γ B is a positive integer. In this situation, Γ B is X-symmetric and all blocks of B are independent sets of Γ (see [1]).Let B be an X-invariant partition of V (Γ). For any vertex σ of Γ, denote by Γ(σ) the set of neighbors of σ in Γ, by Γ B (B) the neighbors of B in Γ B , and by Γ B (σ) the set of blocks of B containing at least one neighbor of σ in Γ. Let b := val(Γ B ), r := |Γ B (σ)|. Then b ≥ r ≥ 1. For any block B of B, denote by v the number of vertices in B, and by Γ(B) the set of vertices of Γ having at least one neighbor in B. For any block C in Γ B (B), denote by Γ[B, C] the bipartite subgraph of Γ induced by (B ∩ Γ(C)) ∪ (C ∩ Γ(B)). Since Γ is X-symmetric, Γ[B, C] is X B∪C -symmetric and is independent of the choice of the adjacent blocks B, C of B up to isomorphism. Let d ≥ 1 be the valency of Γ[B, C] and k := |B ∩ Γ(C)|. Then v ≥ k ≥ d ≥ 1. Again as Γ is X-symmetric, the quintuple p(Γ, X, B) := (v, k, r, b, d) is independent of the choice of σ ∈ V (Γ), B ∈ B and C ∈ Γ B (B). Further assume that B is nontrivial. Then as vr = bk, either v = k ≥ 2 and r = b ≥ 1, or v > k ≥ 1 and b > r ≥ 1. We say Γ is or is not a multicover of Γ B respectively in these cases. The second case happens if and only if the subgraph of Γ induced by vertices from B and C is nonempty and contains at least one isolated vertex.Over the past few decades, finite primitive symmetric graphs have been studied extensively, and several classification results have been achieved based on the O'Nan-Scott theorem [3] and the classification of finite simple groups. In contrast, there are not so many powerful algebraic tools available for dealing with imprimitive symmetric graphs. In general, any imprimitive symmetric graph Γ has a primitive or bi-primitive quotient graph and hence the study of the ...
