Spectral-eccentricity topological index for drug activity prediction

The relationship between molecular topology and biological activity is fundamental to computer-aided drug design. Existing topological indices tend to capture either local connectivity (degree-based indices) or global geometry (eccentricity-based indices), yet relatively few descriptors attempt to fuse both aspects in a unified, per-vertex manner. Here we propose the Spectral Eccentricity Index (SEI), defined as the sum over all vertices of the product of vertex eccentricity and the corresponding Laplacian eigenvalue, where both sequences are sorted in non-decreasing order and paired by rank. This rank-matched pairing, motivated by the rearrangement inequality, amplifies the contribution of vertices that are simultaneously peripherally exposed and spectrally prominent. Rigorous closed-form expressions are derived for path graphs (P_n), cycle graphs (C_n), and complete graphs (K_n). As an exploratory proof of concept, SEI is computed for ten pharmaceutical molecules drawn from the ChEMBL database - caffeine, benzene, naphthalene, adenine, aspirin, glucose, ibuprofen, paracetamol, dopamine, and serotonin - with biological activity (IC₅₀) values against human cyclooxygenase-2 (COX-2, ChEMBL target CHEMBL230). On this limited ten-compound set, SEI yields a coefficient of determination R²=0.923 with measured biological activity, compared with R²=0.781 for the Randíc index and R²=0.717 for the augmented Zagreb index. These figures must be treated strictly as exploratory with only ten data points, a single structurally atypical compound can shift R² by 0.05-0.10, and the apparent advantage of SEI over competing indices cannot be considered established without validation on substantially larger, structurally diverse benchmark sets. All computed SEI values, Laplacian eigenvalues, eccentricity sequences, and the Python script used to reproduce them are provided in the Supplementary Materials to enable independent verification. A per-vertex analysis of aspirin illustrates that SEI places the largest weights on pendant terminal atoms (highest eccentricity, equal to the graph diameter), which correspond to known pharmacophoric groups. With computational complexity O(n³) - tractable for molecules with up to roughly 100 heavy atoms - SEI may serve as a useful spectral-geometric descriptor in QSAR workflows, pending broader validation.