Metric Degree Polynomial and Eccentricity Sequence of Symmetric Hollow Coronoid Polycyclic Conjugated Hydrocarbons

Polycyclic conjugated hydrocarbons (PCHs) with symmetric hollow coronoid structures represent a fundamental class of molecular graphs, exhibiting unique topological and electronic properties. This study systematically investigates the metric degree polynomial and eccentricity sequence for families of PCHs defined by equal arm parameters (t=s=r). Using graph-theoretical and combinatorial methods, we derive explicit formulas and polynomials for both indices, validated by computational approaches for networks of varying sizes. The results reveal regular, scalable patterns in the distribution of eccentricities and metric degrees, reflecting the underlying symmetry and complexity of the networks. We analyze the chemical implications of these topological descriptors, highlighting their value in predicting molecular stability, reactivity, and suitability for advanced materials design. This work not only extends the mathematical foundations of molecular graph theory but also provides practical tools for the rational design of novel organic materials.