Maximum values of the Sombor-index-like graph invariants of trees and connected graphs
Milan Bašić
TL;DR
The paper investigates extremal values of two Sombor-index-like, vertex-degree-based invariants $SO_5$ and $SO_6$ on molecular trees and certain connected graphs. It defines the two invariants via $SO_5(G)=2\pi \sum_{uv\in E(G)} f(d(u),d(v))$ and $SO_6(G)=\pi \sum_{uv\in E(G)} g(d(u),d(v))$ with $f(a,b)=\frac{|a^2-b^2|}{\sqrt{2}+2\sqrt{a^2+b^2}}$ and $g=f^2$, and analyzes their maxima by reducing the problem to degree-pair contributions and univariate optimization. The main contributions include showing that, for molecular trees of order $n$, the maxima are achieved on join graphs $M_{n,k}=\overline{K_{n-k}}\vee K_k$ with explicit expressions $SO_5(M_{n,k})=2\pi k(n-k)f(k,n-1)$ and $SO_6(M_{n,k})=\pi k(n-k)g(k,n-1)$, where the optimal $k$ lies among three integers near $c_0 n$ (with $c_0\approx0.3650$). The paper also extends the discussion to the class of connected graphs, providing an upper bound based on the same join-graph family and conjecturing the bound is tight, highlighting the role of two-value degree sequences and join constructions in these extremal problems. These results advance the understanding of geometry-based degree invariants and their potential chemoinformatic applications.
Abstract
A set of novel vertex-degree-based invariants was introduced by Gutman, denoted by \newline $SO_1, SO_2, \ldots,SO_6$. These invariants were constructed through geometric reasoning based on a new graph invariant framework. Motivated by proposed open problems in [Z. Tang, Q. Li, H. Deng, \textit{Trees with Extremal Values of the Sombor-Index-Like Graph Invariants}, MATCH Commun. Math. Comput. Chem. \textbf{90} (2023) 203-222], we have found the maximum values of $SO_5$ and $SO_6$ in the set of molecular trees with a given number of vertices, respectively, and we have found the maximum value of $SO_5$ in a class of connected graphs.