Geometric approach to degree-based topological index: Elliptic-eccentric Sombor index

The recent emergence of geometric-topological indices as a focus in mathematical chemistry can be traced to Gutman’s introduction of the Sombor index, which prompted the proposal and intensive study of a multitude of indices derived from geometric concepts. The field was significantly advanced in 2024 when Gutman et al. introduced the elliptic Sombor index, followed by Tang et al.’s development of the Euler Sombor index, both of which are based on the fundamental principles of elliptic geometry. Barman and Das further extended this line of research through their hyperbolic Sombor index, derived from hyperbolic eccentricity properties. In this paper, we present the elliptic-eccentric Sombor index (EESO), a novel topological descriptor that expands this important family of geometric indices. Our study makes three contributions to mathematical chemistry: (1) comprehensively establishing the chemical relevance of the EESO index via rigorous QSPR modeling; (2) complete determination of extremal values, including the first three minima and first five maxima across all [Formula: see text]-vertex molecular trees; and (3) full structural characterization of the corresponding extremal graphs. These theoretical advances provide new tools for molecular property prediction and deepen our understanding of structure-property relationships in mathematical chemistry.