Discrimination of Benzenoid Graphs by Topological Polynomials

The discrimination of structural isomers remains a central challenge in mathematical chemistry and molecular graph theory. In this work, twenty topological polynomials are evaluated on benzenoid (polyhex) systems containing between four and seven fused hexagons. Computational analysis of all non-isomorphic benzenoid graphs in this range shows that nine descriptors, including the Theta, Matching, Padmakar–Ivan, Vertex–Szeged, Adjacency Characteristic, Laplacian Characteristic, Signless Laplacian Characteristic, Distance Characteristic, and Harary Matrix Characteristic polynomials, exhibit zero degeneracy and uniquely distinguish all 333 benzenoid isomers with seven hexagons. The structural origin of this behavior is investigated through spectral and distance-based graph properties, and a theoretical framework is proposed distinguishing between (i) computationally verified discrimination on finite sets, (ii) exact algebraic properties and finite-domain completeness for catacondensed benzenoids, and (iii) a composite descriptor completeness conjecture. In contrast, Omega, Sadhana, and Zhang–Zhang polynomials display substantial degeneracy, reflecting information loss associated with quasi-orthogonal cut representations. The obtained results identify a class of topological descriptors with exceptional discriminatory capability and potential applicability in QSAR/QSPR modeling, molecular similarity analysis, and the characterization of extended benzenoid systems.