چکیده انگلیسی:
Suppose $X$ is a simple graph. The $X-$join $Gamma$ of a set of complete or empty graphs ${X_x }_{x in V(X)}$ is a simple graph with the following vertex and edge sets: begin{eqnarray*} V(Gamma) &=& {(x,y) | x in V(X) & y in V(X_x) },\ E(Gamma) &=& { (x,y)(x^prime,y^prime) | xx^prime in E(X) or else x = x^prime & yy^prime in E(X_x)}. end{eqnarray*} The $X-$join graph $Gamma$ is said to be reduced if $x, y in V(X)$, $x ne y$ and $N_X(x) setminus { y} = N_X(y) setminus { x}$ imply that $(i)$ if $xy notin E(X)$ then the graphs $X_x$ or $X_y$ are non-empty; $(ii)$ if $xy in E(X)$ then $X_x$ or $X_y$ 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.
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