Topological Indices in Chemical Graph Theory

Chemical graph theory models molecules as mathematical graphs in which atoms correspond to vertices and bonds to edges. Topological indices are numerical invariants extracted from these graphs that characterise molecular size, branching, connectivity and symmetry. By quantifying these features, such indices correlate with key properties such as boiling and melting points, stability, reactivity and biological activity. The field encompasses degree-based indices (for example the Randić and Zagreb indices), distance-based indices (such as the Wiener index), spectral indices (including the Estrada index) and information-theoretic measures. Recent developments have introduced new descriptors—such as the Sombor index and neighbourhood-degree indices—and have employed generating functions like the M-polynomial to derive families of invariants in a unified way. These tools underpin quantitative structure–property and structure–activity relationships (QSPR/QSAR), facilitate high-throughput screening of drug candidates, inform nanomaterials design and bridge theoretical chemistry with experimental data.

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Topological Indices in Chemical Graph Theory publication trend

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Technical terms

Topological index: A numerical invariant derived from a molecular graph that encodes the connectivity pattern and aids in predicting chemical or physical properties.

Molecular graph: A representation of a molecule in which vertices denote atoms and edges denote chemical bonds.

Vertex degree: The number of edges incident to a vertex, reflecting the bonding environment of an atom in the molecular graph.

Zagreb index: A degree-based index defined as the sum of squared vertex degrees (and its variants), commonly used to quantify molecular branching.

Sombor index: A recent degree-based index calculated as the sum over edges of the square root of the sum of squared degrees of the two adjacent vertices.

Estrada index: A spectral index computed as the sum of exponentials of the eigenvalues of the adjacency matrix, indicative of molecular compactness and cyclicity.

M-polynomial: A generating function that encapsulates contributions of edges classified by degrees, from which multiple topological indices can be derived systematically.

References

1. Spectral analysis of Cupric oxide (CuO) and Graphene Oxide (GO) via machine learning techniques. Egyptian Informatics Journal (2025).
2. Computing Zagreb Indices and Zagreb Polynomials for Symmetrical Nanotubes. Symmetry (2018).
3. QSPR analysis of some novel neighbourhood degree-based topological descriptors. Complex & Intelligent Systems (2021).
4. On Sombor Index. Symmetry (2021).
5. M-Polynomial and Related Topological Indices of Nanostar Dendrimers. Symmetry (2016).