Distinguishing number and distinguishing index of neighbourhood corona of two graphs

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling)  with $d$ labels  that is preserved only by a trivial automorphism. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by  $G_1 \star G_2$  and is the graph obtained by     taking one copy of $G_1$ and  $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\star G_2$. Using results on  automorphisms, we study the distinguishing number and the distinguishing index of $G_1\star G_2$.  We obtain upper bounds for $D(G_1\star G_2)$ and $D'(G_1\star G_2)$.