The Sombor index of trees and unicyclic graphs with given maximum degree

Abstract

Let \(d_G(v)\) be the degree of the vertex \(v\) in a graph \(G\). The Sombor index of \(G\) is defined as \(SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}\), which is a new degree-based topological index introduced by Gutman. Let \(\mathscr{T}_{n,\Delta}\) and \(\mathscr{U}_{n,\Delta}\) be the set of trees and unicyclic graphs with \(n\) vertices and maximum degree \(\Delta\), respectively. In this paper, the tree and the unicyclic graph with minimum Sombor index among \(\mathscr{T}_{n,\Delta}\) and \(\mathscr{U}_{n,\Delta}\) are characterized.