Characterizations of Bounds and Tree Structures for the Hyperbolic Sombor Index
Mitesh J. Patel *
Tolani College of Arts and Science, Adipur - Kachchh, Gujarat, India.
Kajal. S. Baldaniya
KSKV Kachchh University, Bhuj - Kachchh, Gujarat, India.
*Author to whom correspondence should be addressed.
Abstract
For a simple graph G with order n and size m. We study the Hyperbolic Sombor index HSO(G) = \(\sum_{u v \in E} \frac{\sqrt{d_u^2+d_v^2}}{\min \left\{d_u, d_v\right\}}\), where du and dv are the degrees of vertices u and v respectively, a recently introduced degree-based topological index. We first establish several new bounds for HSO(G) in terms of fundamental graph parameters, including the maximum and minimum degrees, the number of pendent vertices, vertex deletion processes and the relationship between a graph and its complement. In addition, we compute exact values of HSO(G) for several well-known families, including Kragujevac trees, perfect binary trees, binary caterpillar trees and dendrimer trees.
Keywords: Hyperbolic sombor index, Kragujevac trees, binary caterpillar trees, dendrimer trees