Symmetry Breaking for Graphs That Are of Importance in Chemistry

The distinguishing number D(G) of a graph G is the least integer ݀ d such that G has a vertex labeling (edge labeling) with ݀ labels that is preserved only by a trivial automorphism. Similar to this definition, the distinguishing index D^ (G) of G is the least integer ݀d such that G has an edge colouring with d colours that is preserved only by a trivial automorphism. Let G be a connected graph constructed from pairwise disjoint connected graphs G1,...,Gk by selecting a vertex of G1 , 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 of G . 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. vertices. Then continue in this manner inductively.