Geometric approach to vertex‐degree‐based topological indices–Elliptic Sombor index, theory and application

Abstract

A novel geometric method is proposed for constructing vertex‐degree‐based molecular structure descriptors (topological indices). The model is based on an ellipse whose focal points represent the degrees of a pair of adjacent vertices. This approach enables a geometric interpretation of several previously known topological indices, and lead to design of a few new. The area of the ellipse induces a vertex‐degree‐based topological index of remarkable simplicity, which we call elliptic Sombor index(). The main mathematical properties of are established, especially its relations to other, earlier known, indices. Then the applicative potential of is analyzed.

Relating Sombor and Euler indices

“…A few years ago, it was discovered that some of these graph invariants have a geometric interpretation (Gutman, 2021). Eventually, this triggered a whole series of geometry-based research studies on VDB invariants (Ali et al, 2024;Gutman, 2022;Gutman et al, 2024;Imran et al, 2022;Liu, 2023b;Tang et al, 2024). The first geometry-motivated VDB invariant is the Sombor index (Gutman, 2021), defined as…”

The Euler Sombor index of a graph

“…Very recently, some geometric methods were proposed for constructing vertex-degree-based molecular structure descriptors (topological indices), the great variety of VDB graph invariants can be constructed by means of geometric arguments, and a few classical VDB topological indices have a geometry-based interpretation, see [8][9][10].…”

“…, and thus induces the first Zagreb index. The area of the ellipse is πr 1 r 2 , induces the elliptic Sombor index [10]…”

“…Its main mathematical properties, especially its relations to other, earlier known, indices, are established and its applicative potential is also analyzed in [10]. In [11] Tang et al determined the maximal value of the elliptic Sombor index of trees with a given diameter or matching number or number of pendent vertices and unicyclic graphs.…”

“…In [10], a novel geometric method, based on an ellipse whose focal points represent the degrees of a pair of adjacent vertices, is proposed for constructing vertex-degree-based molecular structure descriptors. This approach enables a geometric interpretation of several previously known topological indices, and leads to design of a few new.…”