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

Document Type : Research Paper


Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran


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:
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)\}.
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.


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