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

2025-11-25

AI Summary

Determining extremal graphs for degree-based topological indices—particularly for chemical graphs with maximum degree ≤ 3 (i.e., trees, unicyclic, and conjugated molecular graphs)—remains computationally challenging and lacks a unified analytical framework. Method: We propose a polyhedral-theoretic framework that maps degree sequences of such graphs to vertices of convex polyhedra in ℝ³; extremal structures are then identified via linear programming. We implement and publicly release ChemicHull, the first online tool enabling automated search, verification, and visualization of extremal graphs for arbitrary degree-based indices. Contribution/Results: Our approach reveals that only finitely many graph families attain extremal values; it reproduces several classical results, refutes prior conjectures on the extremal graphs for the Randić index, and identifies classes of degree sequences yielding unreachable extrema. The framework unifies the analysis of degree-based index extremality and provides a scalable computational infrastructure for molecular structure optimization and QSPR modeling.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web search models and ranking

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.

Problem

• Determining extremal chemical graphs with maximum degree 3
• Developing online tool for degree-based topological indices analysis
• Identifying counterexamples to published extremal results

Innovation

• Online tool for extremal chemical graphs
• Uses polyhedral framework with linear optimization
• Identifies extremal graphs for topological indices