On the conjecture about the exponential reduced Sombor index

TL;DR

This paper investigates the exponential reduced Sombor index in graphs, specifically characterizing extremal trees with maximal index and disproving a prior conjecture about its bounds.

Contribution

It provides a characterization of extremal trees with the maximum exponential reduced Sombor index and refutes the existing conjecture on its upper bound.

Findings

Extremal trees with maximal exponential reduced Sombor index identified.

The conjecture on the exponential reduced Sombor index is shown to be false.

The result applies to chemical trees of any order n.

Abstract

Let $G=(V(G),E(G))$ be a graph and $d(v)$ be the degree of the vertex $v\in V(G)$. The exponential reduced Sombor index of $G$, denoted by $e^{SO_{red}}(G)$, is defined as $$e^{SO_{red}}(G)=\sum_{uv\in E(G)}e^{\sqrt{(d(u)-1)^2+(d(v)-1)^2}}.$$ We obtain a characterization of extremal trees with the maximal exponential reduced Sombor index among all chemical trees of order $n$. This result shows the conjecture on the exponential reduced Sombor index proposed by Liu, You, Tang and Liu [On the reduced Sombor index and its applications, MATCH Commun. Math. Comput. Chem. 86 (2021) 729--753] is negative.