ChemicHull: an online tool for determining extremal chemical graphs of maximum degree at most 3 for any degree-based topological indices

TL;DR

ChemicHull is an online tool that identifies extremal chemical graphs with maximum degree at most 3 for any degree-based topological index, aiding in chemical graph analysis and property prediction.

Contribution

The paper introduces ChemicHull, a novel online tool that determines extremal chemical graphs for degree-based indices, including the ability to find counterexamples to existing results.

Findings

Successfully recovers known extremal graphs for various indices.

Identifies a counterexample to a previously published extremal result.

Demonstrates effectiveness for chemical trees and unicyclic graphs.

Abstract

Topological indices are graph-theoretic descriptors that play a crucial role in mathematical chemistry, capturing the structural characteristics of molecules and enabling the prediction of their physicochemical properties. A widely studied category of topological indices, known as degree-based topological indices, are calculated as the sum of the weights of a graph's edges, where each edge weight is determined by a formula that depends solely on the degrees of its endpoints. This work focuses exclusively on chemical graphs in which no vertex has a degree greater than 3, a model for conjugated systems. Within a polyhedral framework, each chemical graph is mapped to a point in a three-dimensional space, enabling extremal values of any degree-based topological index to be determined through linear optimization over the corresponding polyhedron. Analysis within this framework reveals that extremality is limited to a small subset of chemical graph families, implying that certain chemical graphs can never attain extremality for any degree-based topological index. The main objective of this paper is to present ChemicHull, an online tool we have developed to determine and display extremal chemical graphs for arbitrary degree-based topological indices. To illustrate the power of this tool, we easily recover established results, emphasizing its effectiveness for chemically significant graph classes such as chemical trees and unicyclic chemical graphs. This tool also enabled the identification of a counterexample to a previously published extremal result concerning the Randi\'c index.