Closed-form connection-number-based topological indices for porphyrazine and tetrakis-porphyrazine networks.

Graph theory offers a mathematical foundation for modeling and analyzing complicated chemical structures and reaction systems. Topological indices are fundamental tools for measuring structural characteristics and predicting physicochemical and biological properties, with broad applications in chemistry, biology, environmental toxicology, and drug discovery. This study focuses on the two molecular graph families, porphyrazine and tetrakis porphyrazine, which contain complex ring structures. For each graph family, we determine the edge partition induced by endpoint connection numbers and use it to derive closed-form expressions for the general Randić connection index, the first and second Zagreb connection indices and coindices, the atom-bond connectivity connection index, the hyper-Zagreb connection index, the geometric-arithmetic connection index, the forgotten connection index, the augmented Zagreb connection index, and three redefined Zagreb connection indices. The dependence of each formula on the graph parameter n is examined in the graphical comparison section. This analysis enables a direct comparison of the growth trends and structural variations of the two graph families.