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

  • Authors:

  • Matt DeVos;Luis Goddyn;Bojan Mohar;Robert Šámal

  • Affiliations:

  • Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada;Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada;Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada;Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

Quantified Score

Hi-index 0.00

Visualization

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 {@l,-@l} 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 {@l,-@l}. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs.