On the distance eccentricity indices of thorn graphs: QSPR and DFT applications to ocular drug design and molecular property prediction

Let G be a connected graph. The distance eccentricity neighborhood of u∈V(G), is defined as N_ε (v)={u∈V(G):d(u,v)=ε(v) }, where ε(u) is the eccentricity of u. The cardinality of deg^{dε}(u)=| N_ε(u)| is called the distance eccentricity degree of the vertex u in G. In 2017, Saleh et al . introduced the first and second distance eccentricity Zagreb indices, defined respectively as the sum of the squares of distance eccentricity degrees and the sum of the products of degrees of adjacent vertices. These indices were left largely unexplored thereafter. This study reexamines and expands upon these indices by introducing several new formulations—such as the third distance eccentricity Zagreb, forgotten, modified forgotten, hyper, Sombor, and sigma indices—and deriving their analytical forms for Thorn graphs, including special subclasses like paths, rods, caterpillars, rings, and stars. To validate their physicochemical relevance, density functional theory (DFT) at the B3LYP/6-311G(d,p) level was employed to calculate thermodynamic and electronic properties of pharmacological agents used in ocular disease treatment, including dipole moments, molecular electrostatic potentials, and highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) energies. Distance-based topological descriptors were then incorporated into quantitative structure–property relationship (QSPR) modeling through curvilinear (quadratic and cubic) regression methods to correlate molecular graph parameters with experimentally measured thermodynamical and physicochemical attributes such as molar volume, polarizability, and refractive index. The cubic regression model exhibited the highest predictive accuracy (R 2 > 0.85), confirming that distance eccentricity-based indices serve as effective topological descriptors capable of bridging molecular structure with thermodynamic and electronic behavior. This integrated framework—linking graph theory, DFT computation, and QSPR analysis—offers a precise and scalable approach for predicting drug performance and advancing the rational design of safer, more efficient therapeutic compounds.