On the first Banhatti-Sombor index

TL;DR

This paper introduces the first Banhatti-Sombor index, a new graph invariant based on vertex degrees, and explores its mathematical properties, relationships with other indices, and extremal trees.

Contribution

It establishes relations between the Banhatti-Sombor index and existing indices, and characterizes extremal trees for this new index.

Findings

Derived mathematical relations with known topological indices.

Characterized extremal trees for the Banhatti-Sombor index.

Provided insights into the index's behavior on chemical trees.

Abstract

Let $d_v$ be the degree of the vertex $v$ in a connected graph $G$. The first Banhatti-Sombor index of $G$ is defined as $BSO(G) =\sum_{uv\in E(G)}\sqrt{\frac{1}{d^2_u}+\frac{1}{d^2_v}}$, which is a new vertex-degree-based topological index introduced by Kulli. In this paper, the mathematical relations between the first Banhatti-Sombor index and some other well-known vertex-degree-based topological indices are established. In addition, the trees extremal with respect to the first Banhatti-Sombor index on trees and chemical trees are characterized, respectively.