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

TL;DR

This paper determines the eigenvalues of (3,6)-fullerenes, a class of cubic plane graphs, proving a conjecture about their spectral symmetry and revealing their structure as Cayley sum graphs.

Contribution

It proves a conjecture on the spectral symmetry of (3,6)-fullerenes and identifies their algebraic structure as Cayley sum graphs using geometric representation.

Findings

Eigenvalues of (3,6)-fullerenes come in symmetric pairs, except for four specific eigenvalues.

(3,6)-fullerenes are examples of Cayley sum graphs.

The spectral structure of these graphs is nearly bipartite.

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 $\{\lambda,-\lambda\}$ 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.