The Distinguishing Number and the Distinguishing Index of Graphs from Primary Subgraphs

The distinguishing number (index)D(G) (D′(G)) of a graph Gis the least integerd such thatG has a vertex labeling (edge labeling) with d labels that is preserved only by the trivial automorphism. LetG be a connected graph constructed from pairwise disjoint connected graphs G1, …, Gkby selecting a vertex ofG1, a vertex of G2, and identifying these two vertices. Then continue in this manner inductively. We say that G is obtained by point-attaching from G1, …, Gk and that Gi’s are the primary subgraphs ofG. In this paper, we consider some particular cases of these graphs that are of importance in chemistry and study their distinguishing number and distinguishing index.