Sombor-index-like invariants of some graphs

TL;DR

This paper introduces and analyzes new Sombor-like graph invariants inspired by geometric considerations, computing their values for specific graph classes such as cactus chains and polymers.

Contribution

It defines new SO-like invariants and evaluates them for various graphs, expanding the understanding of degree-based graph invariants.

Findings

Computed new invariants for specific graph classes

Analyzed relationships between these invariants and classical indices

Provided formulas and properties for the new invariants

Abstract

The Sombor index (SO) is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices of $\sqrt{d_i^2+d_j^2}$, where $d_i$ is the degree of the $i$-th vertex. It has been conceived using geometric considerations. Recently, a series of new SO-like degree-based graph invariants (denoted by $SO_1, SO_2,..., SO_6$) is taken into consideration, when the geometric background of several classical topological indices (Zagreb, Albertson) has considered. In this paper, we compute and study these new indices for some graphs, cactus chains and polymers.