Maximum Diminished Sombor Index of Molecular Trees with a Perfect Matching

TL;DR

The paper studies the diminished Sombor index $DSO(G)=\sum_{uv\in E}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$ for molecular graphs, focusing on molecular trees with a perfect matching to determine the maximum possible value and the extremal structures. It combines an empirical chemical-applicability analysis—showing meaningful correlations between $DSO$ and certain octane-isomer properties—with a rigorous extremal-graph-theory approach that constrains degree-patterns and uses class-based constructions. The main result is a complete classification of extremal trees: for $n\ge12$, the maximum $DSO$ is attained by trees in $\mathcal{G}_0$, $\mathcal{G}_1$, or $\mathcal{G}_2$ corresponding to $n\equiv2,0,4\pmod{6}$, respectively, with a closed-form, piecewise-linear bound in $n$ and explicit equality conditions. These findings provide exact structural descriptions and quantitative benchmarks for the $DSO$ index in chemically relevant molecular-tree graphs, informing both theoretical understanding and potential predictive applications in chemistry.

Abstract

The diminished Sombor index $(DSO)$ of a graph $G$, introduced by Rajathagiri, is defined as $$DSO(G)=\sum_{uv\in E}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$ where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v$. A graph $G$ is a molecular graph if $d_G(u)\leq 4$ for all $u\in V(G)$. In this paper, we examine the chemical applicability of the $DSO$ index for predicting physicochemical properties of octane isomers. We also determine the maximum value of the diminished Sombor index among all molecular trees of order $n$ with perfect matching and characterize all the corresponding extremal trees.

Theorems & Definitions (8)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof