It's an important application for perfect matching counting theories in quantum chemistry

crystal physics and computer science. The research for perfect matching counting theories has a quite important theoretical value and realistic meanings. But the counting problem of perfect matching for general graphs has been proved to be NP—hard. Lovász and Plummer proposed a conjecture on this problem: every 2-edge-connected cubic graph has at least exponential perfect matchings. By applying differentiation

summation and re-nested recursive calculation

several counting formulas of the perfect matching for two specific types of graphs are given. And the results indicate that the conjecture of Lovász and Plummer is true in the case of these two types of graphs.