The distinguishing number and distinguishing index of the lexicographic product of two graphs
Saeid Alikhani, Samaneh Soltani · Discussiones Mathematicae Graph Theory · 2018
The distinguishing number (index) D(G) (D (G)) of a graph G is the least integer d such that G has a vertex labeling (edge labeling) with d labels that is preserved only by the trivial automorphism. The lexicographic product of two graphs G and H, G[H] can be obtained from G by substituting a copy H u of H for every vertex u of G and then joining all vertices of H u with all vertices of H v if uv E(G). In this paper we obtain some sharp bounds for the distinguishing number and the distinguishing index of the lexicographic product of two graphs. As consequences, we prove that if G is a connected graph with Aut(G[G]) = Aut(G)[Aut(G)], then for every natural number k, D(G) D(G k ) D(G) + k -1 and all lexicographic powers of G, G k (k 2) can be distinguished by two edge labels, where