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 $SO(G)={\sum }_{uv\in E(G)}\sqrt{d_G^2(u)+d_G^2(v)}$, which is a new degree-based topological index introduced by Gutman. Let ${\mathcal{T}}_{n,\Delta }$ and ${\mathcal{U}}_{n,\Delta }$ 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 ${\mathcal{T}}_{n,\Delta }$ and ${\mathcal{U}}_{n,\Delta }$ are characterized.

10 pages, 2 figures. arXiv admin note: text overlap with arXiv:2103.04645