distinguishing number and distinguishing index of the join of two graphs

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 gk 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 gk has the distinguishing index 2, except d'(k2 +k2) = 3.