Fullerene graphs with more negative than positive eigenvalues: The exceptions that prove the rule of electron deficiency?

Abstract

In rare cases a neutral fullerene has an antibonding HOMO in simple Hückel theory, thus constituting a counter-example to the mathematical conjecture that in all fullerenes the number of positive eigenvalues of the adjacency matrix is greater than or equal to the number of negatives. The examples found here all have at least 628 carbon atoms and leave undisturbed the rule-of-thumb prediction of electron deficiency for the vast majority of fullerenes.