Nirmala indices bridge molecular connectivity and genetic stability: A graph-theoretic lens on purines and...

Molecular topology lies at the heart of biological complexity and degree-based topological indices serve as fundamental invariants for quantifying the structural and functional organization of biomolecular systems. In this work we present a unified graph-theoretic framework for analyzing the molecular graphs of purines and pyrimidines using the Nirmala index N(G) and its two inverse variants IN₁(G) and IN₂(G). Here G=(V,E) denotes the molecular graph of a purine or pyrimidine molecule, where vertices V represent heavy atoms (C, N, O) and edges E correspond to covalent bonds with hydrogens omitted in accordance with standard chemical graph theory conventions. The five canonical nucleobases-Adenine and Guanine (purines) and Cytosine, Thymine and Uracil (pyrimidines)-are modeled to enable systematic comparison between these two molecular classes. Rigorous degree-based theorems are established to derive extremal bounds for the indices in terms of the minimum and maximum vertex degrees, δ(G) and Δ(G) and to demonstrate analytical duality between IN₁(G) and IN₂(G). Explicit computations reveal that purines exhibit greater degree heterogeneity and larger N(G) values whereas pyrimidines display higher regularity with IN₁(G) and IN₂(G) approaching their theoretical upper bounds. Biologically this degree-sensitive quantification captures structural asymmetry, bonding regularity and intrinsic stability across purine and pyrimidine frameworks. The proposed Nirmala-type indices thus establish a concise mathematical-biological bridge linking molecular topology to functional stability in genetic architectures offering promising applications in cheminformatics, QSAR/QSPR modeling and systems-level analysis of nucleic acids.