Can topological indices transmit information on properties but not on structures?

Area of Science:

• Computational chemistry
• Graph theory
• Cheminformatics

Background:

• Topological indices (TIs) are numerical descriptors of molecular graphs.
• The nature of local vertex invariants (LOVIs) influences TI degeneracy and utility.
• High degeneracy in TIs can complicate structure elucidation via reverse engineering.

Purpose of the Study:

• To survey and categorize topological indices based on LOVI and TI properties.
• To analyze the impact of TI degeneracy on structure-property relationships and reverse engineering.
• To introduce and discuss novel second and third-generation topological indices.

Main Methods:

• Classification of topological indices based on LOVI and TI characteristics (integer/real).
• Analysis of topological index degeneracy and its implications for the inverse problem.
• Introduction and comparative discussion of novel topological indices (G, DN2S(4), TOTOP).

Main Results:

• First-generation TIs with integer LOVIs and TIs exhibit high degeneracy, leading to combinatorial complexity in reverse engineering.
• Low degeneracy TIs may offer property information but hinder structural reverse engineering.
• The second-generation index G shows near-linear correlation with the third-generation triplet TI DN2S(4).

Conclusions:

• Topological index degeneracy is a critical factor in balancing structural information and reverse engineering feasibility.
• Novel topological indices like G, DN2S(4), and TOTOP offer varied levels of degeneracy and correlation.
• The choice of topological index depends on the specific goals of structure-property analysis and reverse engineering.