Abstract
AbstractThe Sanskruti index of a graphGis defined as$$\begin{align*}S(G)=\sum_{uv\in{}E(G)}{\left(\frac{s_G(u)s_G(v)}{s_G(u)+s_G(v)-2}\right)}^3, \end{align*}$$wheresG(u) is the sum of the degrees of the neighbors of a vertexuinG. LetPn,Cn,SnandSn+ebe the path, cycle, star and star plus an edge ofnvertices, 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.In this paper, we investigate the extremal trees and unicyclic graphs with respect to Sanskruti index. More precisely, we show that(1)$\frac{512}{27}n-\frac{172688}{3375}\leq{}S(T)\leq{}\frac{(n-1)^7}{8(n-2)^3}$for ann-vertex treeTwithn≤ 3, with equalities if and only ifT ≌Pn(left) andT≌Sn(right);(2)$ \frac{512}{27}n\leq{}S(G)\leq{}\frac{(n-3)(n+1)^3}{8}+\frac{3(n+1)^6}{8n^3}$for ann-vertex unicyclic graph withn≥ 4, with equalities if and only ifG ≌Cn(left) andG≌Sn+e(right).
Highlights
The Sanskruti index of a graph G is defined as (︂ ∑︁ S(G) = sG(u)sG(v))︂3, uv∈E(G) sG(u) + sG(v) − 2 where sG(u) is the sum of the degrees of the neighbors of a vertex u in G
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
Motivated by the new proposed Sanskruti index, we investigate the extremal trees and extremal unicyclic graphs with respect to this topological index
Summary
)︂3 , uv∈E(G) sG(u) + sG(v) − 2 where sG(u) is the sum of the degrees of the neighbors of a vertex u in G. Topological indices (or molecular structure descriptors) are utilized as a standard tool to study structure-property relations, especially in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) applications [9, 20] These topological indices are studied on chemical graphs, whose vertices correspond to the atoms of molecules and edges correspond to chemical bonds [15, 17,18,19]. Application of topological indices in biology and chemistry began in 1947 with the work of Harold Wiener [20], who introduced the Wiener index to show correlations between physico-chemical properties of organic compounds and the index of their molecular graphs. The main results are proved in Section 3: first, we give lower and upper bounds for the Sanskruti index on trees and provide the extremal graphs (Theorems 9 and 10), we give lower and upper bounds for unicyclic graphs and provide the extremal graphs (Theorems 14 and 15)
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