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Graph theory has various applications in chemistry and molecular structure research, and its importance has been growing continuously. In molecular graphs, points represent the atoms, and lines represent the chemical bonds between atoms. This article is about to show the precise evaluation for topological descriptors based on the degree sum of the neighbors of end vertices for some molecular structures. We derive for Sanskruti index, Neighborhood inverse sum index, Neighborhood Harmonic index, Fifth N D e index, and Neighborhood second modified Zagreb index for two-dimensional lattice structure of certain Nanotube and Nanotorus. [ABSTRACT FROM AUTHOR]

The topological indices are important tools in the chemical graph theory to model different physico-chemical properties of molecules. The Sanskruti index for a graph G having vertex set V(G) and edges set E(G) is defined as S (G) = ∑ u v ∈ E (G) δ G (u) δ G (v) δ G (u) + δ G (v) - 2 3 , where δ G (u) represents the degree sum of nodes adjacent to u ∈ V (G) . In this report, we argue that the chemical applicability of the S -index in the paper (Hosamani in J Appl Math Comput 54:425–433, 2017) is investigated incorrectly. In addition to correcting the result, we perform a comprehensive investigation to uncover the chemical significance of S . Some interesting mathematical features of the index are also explored by deriving some tight bounds with identifying the extremal graphs. [ABSTRACT FROM AUTHOR]

The Sanskruti index of a graph G is defined as S(G)=∑uv∈E(G)sG(u)sG(v)sG(u)+sG(v)−23,$$\begin{align*}S(G)=\sum_{uv\in{}E(G)}{\left(\frac{s_G(u)s_G(v)}{s_G(u)+s_G(v)-2}\right)}^3, \end{align*}$$where sG(u) is the sum of the degrees of the neighbors of a vertex u in G. Let Pn, Cn, Sn and Sn + e be the path, cycle, star and star plus an edge of n vertices, respectively. The Sanskruti index of a molecular graph of a compounds can model the bioactivity of compounds, has a strong correlation with entropy of octane isomers and its prediction power is higher than many existing topological descriptors.

Inspired by recent work of augmented Zagreb index (AZI) we propose here a new topological index, Sanskruti index $$\mathcal {S}(G)$$ of a molecular graph G. In QSAR/QSPR study, topological indices are utilized to guess the bioactivity of chemical compounds. The Sanskruti index $$\mathcal {S}(G)$$ shows good correlation with entropy of an octane isomers. In this paper we compute the Sanskruti index $$\mathcal {S}(G)$$ of line graphs of subdivison graphs of 2D-lattice, nanotube and nanotorus of $$\textit{TUC}_{4}C_{8}[p,q]$$ . [ABSTRACT FROM AUTHOR]