Distinguishing number and distinguishing index of join of two graphs

The distinguishing number (index) $D(G)$ ($D^{\mathrm{\prime }}(G)$) 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. In this paper we study the distinguishing number and the distinguishing index of join of two graphs $G$ and $H$, i.e., $G+H$. We prove that $0\le D(G+H)-max\{D(G),D(H)\}\le z$, where $z$ is depends of the number of some induced subgraphs generated by some suitable partitions of $V(G)$ and $V(H)$. Also, we prove that if $G$ is a connected graph of order $n\ge 2$, then $D^{\mathrm{\prime }}(G+\cdots +G)=2$, except $D^{\mathrm{\prime }}(K_2+K_2)=3$.