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

Suppose X is a simple graph. The X-join \Gamma of a set ofcomplete 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 \inV(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 \not\in 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.

Keywords:

X-join of graphs , reduced X-join of graphs , Automorphism group