[Paper Review] Cayley sum graphs and eigenvalues of $(3,6)$-fullerenes

Matt DeVos, Luis Goddyn+2 more|ArXiv.org|Dec 11, 2007

Graph theory and applications11 references4 citations

TL;DR

This paper proves that the eigenvalues of (3,6)-fullerenes—cubic plane graphs with only triangular and hexagonal faces—come in symmetric pairs {λ, −λ} except for the fixed eigenvalues {3, −1, −1, −1}, confirming a conjecture by Fowler. The authors establish this by showing these graphs are Cayley sum graphs derived from geometric lattices, leveraging group character theory to analyze their spectra.

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 $\{λ,-λ\}$ 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.

Motivation & Objective

• To resolve Fowler's conjecture on the spectral structure of (3,6)-fullerenes, specifically that their eigenvalues are symmetric about zero except for four fixed values.
• To generalize this spectral behavior to a broader class of graphs, including (0,3,6)-fullerenes with semiedges, by introducing a geometric lattice-based construction.
• To establish a connection between geometric crystallographic structures and Cayley sum graphs, enabling spectral analysis through group characters.
• To demonstrate that the spectral symmetry arises from the underlying algebraic and geometric structure of these graphs, not just combinatorial properties.
• To provide a systematic method for computing the spectra of such graphs using lattice quotients and character theory on finite abelian groups.

Proposed method

• Represent (0,3,6)-fullerenes as quotients of a lattice-like graph embedded in the plane, using a geometric construction based on Alexandrov's theorem on polygonal surfaces.
• Define the graphs as Cayley sum graphs over finite abelian groups, where adjacency is determined by u + v ∈ S for u, v in the group Γ.
• Apply character theory to the group Γ: each character χ of Γ yields an eigenvalue χ(S), and the spectrum is determined by real and conjugate pairs of characters.
• Use the decomposition of characters into real-valued (R) and conjugate pairs (C) to show that eigenvalues appear as {χ(S) : χ ∈ R} ∪ {±|χ(S)| : χ ∈ C}, leading to symmetric spectral structure.
• Construct explicit examples using lattices such as D_d and D_d^+ (generalized diamond packing), and embed them via sublattices Λ to generate finite Cayley sum graphs with controlled semiedge counts.
• Demonstrate that when the generating set S is close to 2A, the resulting graph inherits local geometry from the underlying crystallographic tiling, enabling spectral computation.

Experimental results

Research questions

• RQ1 Do all (3,6)-fullerenes have eigenvalues that are symmetric about zero, except for the fixed set {3, −1, −1, −1}?
• RQ2 Can the spectral symmetry observed in (3,6)-fullerenes be extended to a broader class of graphs, including those with semiedges (i.e., (0,3,6)-fullerenes)?
• RQ3 What is the algebraic and geometric structure underlying (3,6)-fullerenes that leads to their spectral symmetry?
• RQ4 How can the spectrum of such graphs be systematically computed using group-theoretic and geometric constructions?
• RQ5 Are there other families of graphs that exhibit 'spectrally nearly bipartite' behavior, i.e., eigenvalues symmetric about zero with only a small fixed set of exceptions?

Key findings

• The spectrum of every (3,6)-fullerene is of the form {3, −1, −1, −1} ∪ L ∪ (−L), where L is a multiset of nonnegative reals, confirming Fowler’s conjecture.
• The authors prove that (0,3,6)-fullerenes also exhibit the same spectral symmetry, extending the result beyond the original conjecture.
• The graphs are shown to be Cayley sum graphs over finite abelian groups, with adjacency defined by u + v ∈ S, enabling spectral analysis via group characters.
• For the D_d-lattice construction with A = (1/2, 0, ..., 0), the resulting Cayley sum graph has exactly 2^d semiedges and spectrum of the form M ∪ L ∪ (−L), where M is {4} or {4, 0}.
• In the diamond-like construction using D_d^+ (e.g., D_3^+ for diamond structure), the spectrum has unmatched eigenvalues M = {4, −2, −2} or M = {4, 0, −2, −2}, depending on lattice choice.
• The 8-dimensional E_8 lattice (D_8^+) and the 24-dimensional Leech lattice yield high-dimensional Cayley sum graphs with spectral symmetry, demonstrating the generality of the framework.