S M Nazmuz Sakib Degree-Spectrum Divergence Principle: A Jensen-Shannon Spectral-Topological Index for Molecular Graphs

Chemical graph theory routinely combines degree-based and spectral descriptors to correlate molecular structure with physicochemical and biological properties. In this manuscript we introduce the S M Nazmuz Sakib Degree-Spectrum Divergence (abbreviated as the Sakib-DSD index), a spectral-topological invariant for simple graphs that measures a Jensen-Shannon divergence between the normalized degree sequence and the normalized squared adjacency spectrum of a graph. This leads to the S M Nazmuz Sakib Degree-Spectrum Divergence Principle: molecular graphs whose local connectivity pattern is strongly mis-aligned with their global spectral energy distribution exhibit larger Sakib-DSD and, in turn, a heightened structural heterogeneity. We derive basic properties of this index, including isomorphism invariance and general bounds, and we introduce the S M Nazmuz Sakib Star-Divergence Constant C Sakib as the limiting value of Sakib-DSD on star graphs. Using explicit numerical computations on classical graph families (paths, cycles, stars) and ensembles of random graphs, we investigate correlations between Sakib-DSD, graph energy, degree variance, and cyclomatic number. We formulate the S M Nazmuz Sakib Degree-Spectrum Divergence Hypothesis, which posits that in Erdős-Rényi-type ensembles with fixed order the expected Sakib-DSD is a monotone increasing function of the cyclomatic number. All figures and illustrations in this work are derived from concrete data obtained by spectral calculations on explicit graphs.