Closed formulas for the number of small paths, independent sets and matchings in fullerenes

Fullerene graphs are 3-regular planar graphs whose faces are pentagons or hexagons. In this paper, we first compute the number of paths of low order in a fullerene. Then apply these numbers to obtain the number of independent sets of size kk and kk-matchings in fullerenes, when k=1,2,3,4k=1,2,3,4.