On the Graovc-Ghorbani and Atom-Bond Connectivity Indices of Graphs from Primary Subgraphs

Let G=(V,E) be a finite simple graph. The Graovac-Ghorbani index of a graph G is defined as ABCGG(G)=∑uv∈E(G) √((nu(uv,G)+nv(uv,G)-2)/(nu(uv,G)nv(uv,G))), where nu(uv,G) is the number of vertices closer to vertex u than vertex v of the edge uv∈E(G).  nv(uv,G) is defined analogously. The atom-bond connectivity index of a graph G is defined as ABC(G)=∑uv∈E(G)√((du+dv-2)(dudv)), where du is the degree of vertex u in G. 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 give some lower and upper bounds on Graovac-Ghorbani and atom-bond connectivity indices for these graphs. Additionally, we consider some particular cases of these graphs that are of importance in chemistry and study their Graovac-Ghorbani and atom-bond connectivity indices.