On the Graovac-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 $ABC_{GG}(G)=\sum_{uv\in E(G)}\sqrt{\frac{n_u(uv,G)+n_v(uv,G)-2}{n_u(uv,G)n_v(uv,G)}},$where$n_u(uv,G)$isthenumberofverticesclosertovertex$u$thanvertex$v$ of the edge $uv\in E(G)$. $n_v(uv,G)$ is defined analogously. The atom-bond connectivity index of a graph G is defined as $ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_ud_v}},$where$d_u$isthedegreeofvertex$u$in$G$\mathrm{.}Let$G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\dots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identifying these two vertices. Then continue in this manner inductively. We say that $G$ is obtained by point-attaching from $G_1, \ldots ,G_k$andthat$G_i${}^{\mathrm{\prime }}saretheprimarysubgraphsof$G$. 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.