Nonclassical Cn (n = 30-40, 50) fullerenes containing five-, six-, seven-member rings

► Classical and heptagon-contained nonclassical fullerene isomers of C30-C40 and C50 are studied. ► Pentagon adjacency penalty rule (PAPR) is roughly applicable for nonclassical fullerenes with heptagons. ► Hexagon neighbor rule (HNR) almost breaks down for the heptagon-contained nonclassical fullerenes. ► A new topological index U with better performance over PAPR and HNR is presented.