Publication at Faculty of Mathematics and Physics |
2009
Abstract
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 {c, -c} 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 {c, -c}.
Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs.