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<p>The max–min degree index is one of the most important topological indices among the defined 148 discrete Adriatic indices (Vukicević and Gasperov in Adriatic Indices Croat Chem Acta 83(3):243–260, 2010). Vukičević proposed some problems related to the upper and lower bounds of the max–min degree index of a graph. Recently, Das et al. characterized the graphs’ extremal concerning the max–min degree index over connected graphs, trees, and unicyclic graphs with a given number of vertices. Here we determine the <i>n</i>-vertex trees with minimum for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ge 3\)</EquationSource> </InlineEquation>, the second, the third and the fourth for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\ge 7\)</EquationSource> </InlineEquation> and fifth for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n\ge 10\)</EquationSource> </InlineEquation> minimum <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(Mm_{deg}\)</EquationSource> </InlineEquation> indices, unicyclic graphs with the minimum, the second minimum, the third minimum and the fourth minimum <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(Mm_{deg}\)</EquationSource> </InlineEquation> indices, and bicyclic graphs with the minimum for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(n\ge 4\)</EquationSource> </InlineEquation>, the second for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(n\ge 6\)</EquationSource> </InlineEquation> and the third for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(n\ge 7\)</EquationSource> </InlineEquation> minimum <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(Mm_{deg}\)</EquationSource> </InlineEquation> indices. Finally, we characterize the extremal graphs concerning the max–min degree index over chemical graphs, chemical trees, and obtain a relation between the energy of a graph and the max–min degree index.</p>

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Max–Min Degree Index of Trees, Unicyclic, Bicyclic and Chemical Graphs

  • 期刊论文

  • 发表日期：

    2025年12月6日

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  • Biswaranjan Khanra,
  • Shibsankar Das

摘要

The max–min degree index is one of the most important topological indices among the defined 148 discrete Adriatic indices (Vukicević and Gasperov in Adriatic Indices Croat Chem Acta 83(3):243–260, 2010). Vukičević proposed some problems related to the upper and lower bounds of the max–min degree index of a graph. Recently, Das et al. characterized the graphs’ extremal concerning the max–min degree index over connected graphs, trees, and unicyclic graphs with a given number of vertices. Here we determine the n-vertex trees with minimum for \(n\ge 3\) , the second, the third and the fourth for \(n\ge 7\) and fifth for \(n\ge 10\) minimum \(Mm_{deg}\) indices, unicyclic graphs with the minimum, the second minimum, the third minimum and the fourth minimum \(Mm_{deg}\) indices, and bicyclic graphs with the minimum for \(n\ge 4\) , the second for \(n\ge 6\) and the third for \(n\ge 7\) minimum \(Mm_{deg}\) indices. Finally, we characterize the extremal graphs concerning the max–min degree index over chemical graphs, chemical trees, and obtain a relation between the energy of a graph and the max–min degree index.