Distinguishing number and distinguishing index of Kronecker product 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 Kronecker product $G\times H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\times V (H)$ and edge set $\{\{(u, x), (v, y)\} | \{u, v\} \in E(G) ~and ~\{x, y\} \in E(H)\}$. In this paper we study the distinguishing number and the distinguishing index of Kronecker product of two graphs.