ChemicHull: an online tool for determining extremal chemical graphs of maximum degree at most 3 for any degree-based topological indices
Sébastien Bonte, Gauvain Devillez, Valentin Dusollier, Alain Hertz, Hadrien Mélot, David Schindl
TL;DR
The paper develops a polyhedral framework for extremal degree-based topological indices on chemical graphs with maximum degree $3$, encoding each graph by the triple $(m_{12},m_{13},m_{33})$ and reducing optimization to linear programming over the polytope $\mathcal{P}_{n,m}$. It proves that the extreme points of these polytopes are drawn from a small catalog (at most 16 points, including a parity-dependent $\mathrm{V1}$), enabling exact identification of extremal graphs for any index via evaluation on these points. The authors introduce ChemicHull, an online tool that visualizes $\mathcal{P}_{n,m}$, lists extreme points/facets, and interactively displays graphs achieving the optima for chosen indices, including reproducing known results and correcting prior claims (e.g., a counterexample to a Randić-extremal result). The framework is shown to capture and generalize results for trees, unicyclic, and bicyclic classes, with concrete conditions for when paths or cycles are optimal and with specific maximizers/minimizers for indices like $R_{-1}$, $R_{-1/2}$, $SO$, $SO_{red}$, and $ABS$, offering a practical, scalable approach for extremal graph theory in chemical graph domains.
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 Randić index.