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<p>A topological index is a numerical representation of a graph’s structure that provides significant information pertaining to the graph. Topological indices, similar to the mathematical modeling of the toxicological, pharmacological, physicochemical, and other characteristics of chemical compounds, are extensively utilized in chemical research. Numerous topological indices have been defined in the literature according to degree, distance, spectrum etc. We examine the Euler Sombor index which is defined as <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2506_Article_Equa.gif" Format="GIF" Height="49" Rendition="HTML" Resolution="72" Type="Linedraw" Width="386" /> </MediaObject> <EquationSource Format="TEX">\(EP(G) = \sum\limits_{uv \in E(G)} \sqrt {{{\left( {{d_G}(u)} \right)}^2} + {{\left( {{d_G}(v)} \right)}^2} + {d_G}(u){d_G}(v)},\)</EquationSource> </Equation></p><p>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2506_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\({d_G}(u)\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2506_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\({d_G}(v)\)</EquationSource> </InlineEquation> denote the degrees of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2506_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(u\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2506_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(v\)</EquationSource> </InlineEquation>, respectively. In this paper, we determine the extremal bounds for the Euler Sombor index for molecular trees when the number of vertices and the number of pendent vertices are given. We analyze the correlation for the Euler Sombor index with other established indices utilizing linear, polynomial, ridge, lasso, elastic net, exponential, and logarithmic regressions. Furthermore, we examine the chemical relevance of the Euler Sombor index concerning the isomers of octane.</p>

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On the bounds for the Euler Sombor index of molecular trees and its applications

  • 期刊论文

  • 发表日期：

    2025年4月29日

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  • Qing Zhang,
  • Fairouz Tchier,
  • Zaryab Hussain,
  • Adnan Aslam,
  • Xuewu Zuo

摘要

A topological index is a numerical representation of a graph’s structure that provides significant information pertaining to the graph. Topological indices, similar to the mathematical modeling of the toxicological, pharmacological, physicochemical, and other characteristics of chemical compounds, are extensively utilized in chemical research. Numerous topological indices have been defined in the literature according to degree, distance, spectrum etc. We examine the Euler Sombor index which is defined as \(EP(G) = \sum\limits_{uv \in E(G)} \sqrt {{{\left( {{d_G}(u)} \right)}^2} + {{\left( {{d_G}(v)} \right)}^2} + {d_G}(u){d_G}(v)},\)

where \({d_G}(u)\) and \({d_G}(v)\) denote the degrees of \(u\) and \(v\) , respectively. In this paper, we determine the extremal bounds for the Euler Sombor index for molecular trees when the number of vertices and the number of pendent vertices are given. We analyze the correlation for the Euler Sombor index with other established indices utilizing linear, polynomial, ridge, lasso, elastic net, exponential, and logarithmic regressions. Furthermore, we examine the chemical relevance of the Euler Sombor index concerning the isomers of octane.