On the conjecture about the exponential reduced Sombor index

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.