### 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 λ₁ ≥ λ₂ ≥ ∙∙∙ ≥ λ_{n}. It is known that the largest eigenvalue is 3. Let*R*= {f_{i}|*i*ϵ Z_{l}} be a set of*l*faces of a fullerene*F*such that*f*is adjacent to_{i}*f*_{i}₊₁,*i*ϵ Z_{l}, via an edge e_{i}. If the edges in { e_{i}|*i*ϵ 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 2*r*─ 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. Copyright © 2010 American Scholars Press Inc.Original language | English |
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Pages (from-to) | 41-51 |

Journal | Australasian Journal of Combinatorics |

Volume | 47 |

Publication status | Published - Jun 2010 |