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

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

2007-12-11

arxiv.org/abs/0712.1631v2

Mathematics

Combinatorics

Scientific paper

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 $\{\lambda,-\lambda\}$ except for the four eigenvalues $\{3,-1,-1,-1\}$. We exhibit other families of graphs which are "spectrally nearly bipartite" in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs.

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