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

TL;DR

This paper characterizes the extremal chemical graphs with maximum degree at most 3 for degree-based topological indices by describing a polytope with at most 10 facets and at most 16 extreme points, simplifying their analysis.

Contribution

It provides a complete polyhedral description of extremal chemical graphs with degree constraints, reducing the problem to analyzing a small set of extreme points.

Findings

Maximum of 16 extremal graphs for given parameters

Polytope with at most 10 facets describes all extremal cases

Simplifies the search for extremal chemical graphs

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.