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Fullerene Graph, Tubular Fullerene, Perfect Matching, Cyclic Edge-cut
The perfect matchings counting problem of graphs has important applications in combinatorial optimization, statistical physics, quantum chemistry and other fields. A perfect matching of a graph G is a set of non-adjacent edges that covers all vertices of G . The number of perfect matchings of a graph is closely related to its number of vertices. A fullerene graph is a 3-connected cubic planar graphs all of whose faces are pentagons and hexagons. Došlić obtained that a fullerene graph with P vertices has at least P/2+1 perfect matchings, Zhang et al. proved a better lower bound 3(p+2)/4 of the number of perfect matchings of a fullerene graph. We have known that the fullerene graph has a nontrivial cyclic 5-edge-cut if and only if it is isomorphic to the graph Tn for some integer n >=1, where Tn is the tubular fullerene graph Tn comprised of two caps formed of six pentagons joined by n concentric layers of hexagons. In this paper, the perfect matchings of the graph Tn is classified by matching a certain vertex, and recursive relations of a set of perfect matching numbers are obtained. Then the calculation formula of the number of perfect matchings of the graph Tn is given by recursive relationships. Finally, we get the number of perfect matchings of Tn with P vertices.
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