Abstract
A new type of vertex-degree-based topological indices of a graph, called the reduced Sombor index, is proposed by Gutman very recently. Accurately, for a graph \(G=(V(G),E(G))\), the reduced Sombor index of \(G\), denoted by \(SO_{red}(G)\), is defined
as \[ SO_{red}(G)=\sum\limits_{uv\in E(G)}\sqrt{(d(u)-1)^2+(d(v)-1)^2}, \] where \(d(u)\) denotes the degree of the vertex \(u\) in \(G\). In this note, we fix a flaw of a theorem on the upper bound for the reduced Sombor index of a bipartite
graph and prove that TurĂ¡n graph has the maximum reduced Sombor index among all \(k\)-chromatic graphs, which solves a conjecture on the 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).