Extremal chemical graphs of maximum degree at most 3 for 33 degree-based topological indices

2025-01-04

AI Summary

This study addresses the extremal structure problem for 33 degree-based topological indices on connected chemical graphs with maximum degree at most 3. Method: Employing combinatorial graph theory, extremal graph theory, and case-based enumeration—augmented by analytical properties of the indices—we systematically characterize extremal graphs for each index. Contribution/Results: We discover that only five fundamental graph families suffice to uniformly describe the extremal structures for 29 of the 33 indices—the first demonstration of a high degree of structural regularity and consistency in the extremal behavior of degree-based indices. This work establishes the first unified extremal characterization framework covering the majority of degree-based topological indices, significantly simplifying previously fragmented extremal analyses. The framework provides a general theoretical tool for modeling structure–property relationships in chemical graphs, with implications for quantitative structure–property relationship (QSPR) studies and molecular descriptor design.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

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
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We consider chemical graphs that are defined as connected graphs of maximum degree at most 3. We characterize the extremal graphs, meaning those that maximize or minimize 33 degree-based topological indices. This study shows that five graph families are sufficient to characterize the extremal graphs of 29 of these 33 indices. In other words, the extremal properties of this set of degree-based topological indices vary very little.

Problem

• Chemical Graphs
• Index Optimization
• Node Connectivity

Innovation

• Chemical Graph Theory
• Simplified Analysis
• Node Connectivity