An independent set of a graph G is a set of vertices of G that are pairwise non-adjacent. The independence number, alpha(G), is the order of a maximum independent set of G. A survey of independence numbers is presented for the set of over 10 million fullerene isomers from 20 to 120 carbon atoms and comparisons are made with the pentagon adjacency count, as a measure of thermodynamic stability. Contrary to a literature proposal, stability and minimal independence number of fullerenes are poorly correlated. (c) 2007 Elsevier B.V. All rights reserved.