University of KashanMathematics Interdisciplinary Research2538-36394220191201Distinguishing Number and Distinguishing Index of the Join of Two Graphs23925110210910.22052/mir.2020.133523.1102ENSaeid AlikhaniDepartment of Mathematics, Yazd University, Yazd, IranSamaneh SoltaniDepartment of Mathematics, Yazd University, Yazd, IranJournal Article20180526The 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. In this paper we study the distinguishing number and the distinguishing index of the join of two graphs G and H, i.e., G+H. We prove that 0≤ D(G+H)-max{D(G),D(H)}≤ z, where z depends on the number of some induced subgraphs generated by some suitable partitions of V(G) and V(H). Let G<sup>k</sup> be the k-th power of G with respect to the join product. We prove that if G is a connected graph of order n ≥ 2, then G<sup>k</sup> has the distinguishing index 2, except D'(K<sub>2</sub>+K<sub>2</sub>)=3.https://mir.kashanu.ac.ir/article_102109_ff92223a27f0fd1dfbcd2f75fc2bc091.pdf