The automorphism group of the reduced complete-empty X− join of graphs

Suppose X is a simple graph. The X−join Γ of a set of complete or empty graphs {Xx}x∈V(X) is a simple graph with the following vertex and edge sets: V(Γ)E(Γ)=={(x,y) | x∈V(X) & y∈V(Xx)},{(x,y)(x′,y′) | xx′∈E(X) or else x=x′ & yy′∈E(Xx)}. The X−join graph Γ is said to be reduced if x,y∈V(X), x≠y and NX(x)∖{y}=NX(y)∖{x} imply that (i) if xy∉E(X) then the graphs Xx or Xy are non-empty; (ii) if xy∈E(X) then Xx or Xy are not complete graphs. The aim of this paper is to explore how the graph theoretical properties of X−join of graphs effect on its automorphism group. Among other results we compute the automorphism group of reduced complete-empty X−join of graphs.