Monomer–dimer model on a scale-free small-world network

The explicit determination of the number of monomer–dimer arrangements on a network is a theoretical challenge, and exact solutions to monomer–dimer problem are available only for few limiting graphs with a single monomer on the boundary, e.g., rectangular lattice and quartic lattice; however, analytical research (even numerical result) for monomer–dimer problem on scale-free small-world networks is still missing despite the fact that a vast variety of real systems display simultaneously scale-free and small-world structures. In this paper, we address the monomer–dimer problem defined on a scale-free small-world network and obtain the exact formula for the number of all possible monomer–dimer arrangements on the network, based on which we also determine the asymptotic growth constant of the number of monomer–dimer arrangements in the network. We show that the obtained asymptotic growth constant is much less than its counterparts corresponding to two-dimensional lattice and Sierpinski fractal having the same average degree as the studied network, which indicates from another aspect that scale-free networks have a fundamentally distinct architecture as opposed to regular lattices and fractals without power-law behavior.