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 M1*()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.