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

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