On lower bounds on the number of perfect matchings in n-extendable bricks

Using elements of the structural theory of matchings and a recently proved conjecture concerning bricks, it is shown that every n-extendable brick (except K4, C6 and the Petersen graph) with p vertices and q edges contains at least q − p + (n − 1)!! perfect matchings. If the girth of such an n-extendable brick is at least five, then this graph has at least q − p + nn−1 perfect matchings. As a consequence, the best currently known lower bound on the number of perfect matchings in a fullerene graph is obtained.