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

2021/09/28 by Ghanbari, Nima · 1 citation
#05C09 #05C12 #05C92 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.13564

Abstract

Let G=(V,E) be a finite simple graph. The Graovac-Ghorbani index of a graph G is defined as ABC_GG(G)=∑_{uv∈ E(G)}√((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∈ E(G). n_v(uv,G) is defined analogously. The atom-bond connectivity index of a graph G is defined as ABC(G)=∑_{uv∈ E(G)}√((d_u+d_v-2)/(d_ud_v)), where d_u is the degree of vertex u in G. Let G be a connected graph constructed from pairwise disjoint connected graphs G₁,… ,G_k 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, … ,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.

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• Computing fault-tolerant metric dimension of graphs using their primary subgraphs

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