Complete polyhedral description of chemical graphs of maximum degree at most 3

2025-06-24

AI Summary

This study addresses the extremal structure characterization of chemical graphs with maximum degree at most 3, under fixed numbers of vertices (n) and edges (m), for degree-based topological indices (e.g., Wiener, Zagreb). Methodologically, it reformulates the graph-theoretic extremal problem as a linear program over a low-dimensional polyhedron—specifically, one with at most ten facets—and proves that its extreme points are bounded by 16. The analysis integrates graph theory, combinatorial optimization, and polyhedral theory to derive a general solution for extremizing edge-weighted sums under degree-sequence constraints. Contributions include: (i) a complete classification of all possible extremal graph structures into at most 16 canonical types; (ii) demonstration that, for any (n,m), an extremal chemical graph must belong to one of these types—thereby drastically reducing the search space for molecular structure optimization; and (iii) revelation of high structural consistency across extremal graphs for diverse topological indices, providing a unified theoretical foundation for quantitative structure–property relationship modeling in cheminformatics.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for 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

Chemical graphs are simple undirected connected graphs, where vertices represent atoms in a molecule and edges represent chemical bonds. A degree-based topological index is a molecular descriptor used to study specific physicochemical properties of molecules. Such an index is computed from the sum of the weights of the edges of a chemical graph, each edge having a weight defined by a formula that depends only on the degrees of its endpoints. Given any degree-based topological index and given two integers $n$ and $m$, we are interested in determining chemical graphs of order $n$ and size $m$ that maximize or minimize the index. Focusing on chemical graphs with maximum degree at most 3, we show that this reduces to determining the extreme points of a polytope that contains at most 10 facets. We also show that the number of extreme points is at most 16, which means that for any given $n$ and $m$, there are very few different classes of extremal graphs, independently of the chosen degree-based topological index.

Problem

• Characterize chemical graphs with max degree ≤3 polyhedrally
• Determine extremal graphs optimizing topological indices
• Bound polytope facets and extreme points for graphs

Innovation

• Polyhedral description for chemical graphs
• Degree-based topological index optimization
• Extreme points of polytope analysis