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

TL;DR

This work provides a complete polyhedral description for chemical graphs with maximum degree at most $3$, recasting the extremal problem for any degree-based topological index into a linear optimization over a 3D polytope $\,\mathcal{P}_{n,m}$ spanned by $(m_{12},m_{13},m_{33})$. By exhaustively characterizing 96 $(n,m)$ cases, it shows each feasible polytope has at most 10 facets and at most 16 extreme points, implying only a small set of graph families can realize extremal indices. The paper enumerates 21 facet-defining inequalities and 21 realizable points, classifies 75 polytope types (plus 11 new ones for small $(n,m)$), and demonstrates practical use via an Albertson index example. It further discusses the extension to degree up to 4, noting a substantial combinatorial blow-up, and provides a web resource (ChemicHull) for interactive exploration of the polytopes and extreme points. These results offer a rigorous, scalable framework to identify extremal chemical graphs across a broad class of indices.

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.

Figures (5)

• Figure 1: Two non-isomorphic chemical graphs with coordinates $(1,0,0)$ in $\mathcal{P}_{6,6}$
• Figure 2: Some special graphs with their coordinates in a degenerated polytopes
• Figure 3: The three chemical graphs of order 5 and size 6, with their coordinates in $\mathcal{P}_{5,6}$
• Figure 4: A chemical graph which shows that $(0,0,0)$ is realizable for $(n,m)=(20,23)$
• Figure 5: Two chemical graphs that minimize the Albertson index for $(n,m)=(13,15)$

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Theorem 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof