AbstractThe Randić (connectivity) index is one of the most successful molecular descriptors in structure‐property and structure‐activity relationships studies. We determine the n‐vertex trees with the third for n ≥ 7, the fourth for n ≥ 10, the fifth and the sixth for n ≥ 11 maximum Randić indices, unicyclic graphs with the third for n ≥ 5, the fourth for n ≥ 7, and the fifth for n ≥ 8 maximum Randić indices, and bicyclic graphs with the second and the third for n ≥ 7, the fourth and the fifth for n ≥ 9 maximum Randić indices. We also determine the n‐vertex tree(s) for n ≥ 6 with the fifth minimum Randić index, unicyclic graphs for n ≥ 5 with the second, the third and the fourth minimum Randić indices, and bicyclic graph(s) for n ≥ 5 with the second minimum Randić index. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011