Conjugated open-ended cones in which the configured pentagons are consistent, lies in the circle of Fries Kekule structure. This nonadjacent tightest configuration of pentagons as shown in Figure 1 is consistent with a Fries Kekule structure and thus provides the most stable cone. In the study, various topological indices of the same structure in regard of physico-concoction resources and bioactivity of substance mixtures are studied which further helps to study the behavior of chemical compounds. In this regard, Zagreb indices M-1*(G) and M2*(G) of a molecular graph G is used to evaluate the complexity in chemical systems and biological organisms. In this article, we consider two complex families of stable carbon nanocones and compute their ECI, TEI, and eccentricity-based Zagreb indices.