%0 Journal Article
%T Distinguishing Number and Distinguishing Index of the Join of Two Graphs
%J Mathematics Interdisciplinary Research
%I University of Kashan
%Z 2538-3639
%A Alikhani, Saeid
%A Soltani, Samaneh
%D 2019
%\ 12/01/2019
%V 4
%N 2
%P 239-251
%! Distinguishing Number and Distinguishing Index of the Join of Two Graphs
%K Distinguishing index
%K distinguishing number
%K join
%R 10.22052/mir.2020.133523.1102
%X 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.
%U https://mir.kashanu.ac.ir/article_102109_ff92223a27f0fd1dfbcd2f75fc2bc091.pdf