Presentation on YouTube.

Graphs provide a visual representation of phenomena we encounter every day. Drawing a graph is a way of modeling a potentially complex situation using points and lines, which allows us to eliminate unnecessary elements and focus on the essentials. I'll illustrate this with graph chemistry, also known as topological chemistry, which is the application of graph theory to model the structures of molecules.

A molecule is a group of two or more atoms, of the same or different types, linked together by 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 bonds of a molecule, each bond 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 molecules with n atoms and m bonds that maximize or minimize a given degree-based topological index. Focusing on conjugated systems, such as alkenes, polyenes, benzenoids, and fullerenes, where each atom participates in at most three single bonds, we show that this problem 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 molecules, independently of the chosen degree-based topological index.