Complexity of molecules

Area of Science:

• Computational chemistry
• Cheminformatics
• Graph theory

Background:

• Existing molecular complexity measures, such as those by Bertz and Randic or spanning tree-based methods, are widely used.
• There is a continuous need for refined and alternative methods to quantify molecular complexity.

Purpose of the Study:

• To review established molecular complexity measures.
• To propose novel molecular complexity measures based on vertex and edge weights.
• To compare the proposed measures with existing ones using selected molecular graphs.

Main Methods:

• Review of existing molecular complexity indices (Bertz, Randic, spanning trees).
• Definition of new complexity measures: sum of vertex-weights (squares of vertex-degrees) and sum of edge-weights (products of vertex-degrees).
• Analysis of measures considering partitions of weights and their behavior with molecular size, cyclicity, branching, and symmetry.

Main Results:

• The proposed vertex-weight and edge-weight measures, along with their variants, were defined.
• All reviewed and proposed indices generally increase with molecular size, cyclicity, and branching.
• A key difference emerged in how the measures handle molecular symmetry.

Conclusions:

• The novel vertex- and edge-weight-based complexity measures provide valuable insights into molecular structure.
• These new measures offer a different perspective on molecular complexity, especially regarding the impact of symmetry.
• The study contributes to the development of quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies.