Cayley sum graphs and eigenvalues of (3,6)-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.