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We determine the spectra of cubic plane graphs whose faces have sizes 3 and6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who areinterested in their energy spectra. In particular we prove a conjecture ofFowler, which asserts that all their eigenvalues come in pairs of the form$\{\lambda,-\lambda\}$ except for the four eigenvalues $\{3,-1,-1,-1\}$. Weexhibit other families of graphs which are "spectrally nearly bipartite" inthis sense. Our proof utilizes a geometric representation to recognize thealgebraic structure of these graphs, which turn out to be examples of Cayleysum graphs.

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