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) = {(x, y) | x ∈ v (x) & y ∈ v (xx)}, e(γ) = {(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.