Pfaffian orientations and perfect matchings of scale-free networks

Counting perfect matchings is of great interest in theoretical computer science, mathematics, among other disciplines. However, explicitly determining the number of perfect matchings of general graphs is a challenge. In this paper, we present a first analytical study of the enumeration problem of perfect matchings on scale-free networks, which are ubiquitous in real-life systems. We focus on two iteratively growing scale-free networks: one is fractal, while the other is non-fractal. For each network, we construct an orientation and prove that it is Pfaffian. Then, based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a Pfaffian orientation, we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations. Furthermore, using the recursive relations, we show that for both scale-free networks, their entropy is zero, which is in sharp contrast to the previously obtained results for other networks without scale-free behavior, but having the same average degree as the studied networks, such as square lattice and honeycomb lattice, whose entropies are larger than zero. Our results indicate from another angle that scale-free networks have a distinct architecture in comparison with those lattices lacking the power-law property.