Cayley sum graphs and eigenvalues of -fullerenes

We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, “ ( 3 , 6 ) -fullerenes,” have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form { λ , − λ } except for the four eigenvalues { 3 , − 1 , − 1 , − 1 } . We exhibit other families of graphs which are “spectrally nearly bipartite” in the sense that nearly all of their eigenvalues come in pairs { λ , − λ } . Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs.