On the Graovac-Ghorbani and atom-bond connectivity indices of graphs from primary subgraphs

Abstract

Let $G=(V,E)$ be a finite simple graph. The Graovac-Ghorbani index of a graph G is defined as $AB{C}_{GG}(G)={\sum }_{uv\in E(G)}\sqrt{\frac{n_u(uv,G)+n_v(uv,G)-2}{n_u(uv,G)n_v(uv,G)}},$ where $n_u(uv,G)$ is the number of vertices closer to vertex $u$ than vertex $v$ 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 $ABC(G)={\sum }_{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_u{d}_v}},$ where $d_u$ is the degree of vertex $u$ in $G$. 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 $G_1,\dots ,G_k$ and that $G_i$'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.

31 pages, 23 figures. arXiv admin note: substantial text overlap with arXiv:2106.06562, arXiv:2103.13663

References in corpus (2)

• Remarks on the Graovac-Ghorbani index of bipartite graphs
• Sombor Index of Polymers