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 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.