On the spectra of the fullerenes that contain a nontrivial cyclic-5-cutset

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 ≥ · · · ≥ λn. It is known that the largest eigenvalue is 3. Let R = {fi | i ∈ Zl} be a set of l faces of a fullerene F such that fi is adjacent to fi+1, i ∈ Zl, via an edge ei. If the edges in {ei | i ∈ Zl} 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 ∗ This work is partially supported by GRF, Research Grant Council of Hong Kong; FRG, Hong Kong Baptist University. † Corresponding author 42 W.C. SHIU, W. LI AND W.H. CHAN 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 λr+1 ≥ 1.