The Sombor index and coindex of two-trees

TL;DR

This paper investigates the maximum, second maximum, minimum, and second minimum values of the Sombor index and coindex specifically for two-trees, providing new bounds and characterizations.

Contribution

It determines the extremal values of the Sombor index and coindex for two-trees, a specific class of graphs, which was not previously explored.

Findings

Identified maximum and second maximum Sombor index in two-trees.

Determined minimum and second minimum Sombor coindex in two-trees.

Provided bounds and characterizations for these indices.

Abstract

The Sombor index of a graph $G$, introduced by Ivan Gutman, is defined as the sum of the weights $\sqrt{d_G(u)^2+d_G(v)^2}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of vertex $u$ in $G$. The Sombor coindex is recently defined as $\bar{SO}(G)=\sum \limits_{uv\notin E(G)}\sqrt{d_G(u)^2+d_G(v)^2}$. In this paper, the maximum and second maximum Sombor index, the minimum and second minimum Sombor coindex in two-trees are determined.