Document Type : Original Scientific Paper

**Authors**

Department of Mathematics, Yazd University, Yazd, Iran

**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. 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^{k} 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^{k} has the distinguishing index 2, except D'(K_2+K_2)=3.

**Keywords**

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Autumn 2019

Pages 239-251