## Abstract

A fullerene, which is a 3-connected cubic plane graph whose faces are pentagons and hexagons, is cyclically 5 edge-connected. For a fullerene of order n, it is customary to index the eigenvalues in non-increasing order λ _{1} ≥ λ _{2} ≥ · · · ≥ λ _{n}. It is known that the largest eigenvalue is 3. Let R = {f _{i}|i∈ ℤ _{l}} be a set of l faces of a fullerene F such that f _{i} is adjacent to f _{i+1}, i ∈ ℤ, via an edge e _{i}. If the edges in {e _{i} | i ∈ ℤ _{l}} are independent, then we say that R forms a ring of l faces. In this paper we show that if a fullerene contains a nontrivial cyclic-5-cutset, then it has 2r - 2 eigenvalues that can be arranged in pairs {μ,-μ} (1 < μ < 3), where r is the number of the rings of five faces. Meanwhile 1 is one of its eigenvalues and λ _{r+1} ≥ 1.

Original language | English |
---|---|

Pages (from-to) | 41-51 |

Number of pages | 11 |

Journal | Australasian Journal of Combinatorics |

Volume | 47 |

Publication status | Published - Jun 2010 |

## Scopus Subject Areas

- Discrete Mathematics and Combinatorics