The Distinguishing Number and Distinguishing Index of the Lexicographic Product of Two Graphs

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 G^k = G[G[. . . ]].