  We consider the disordered monomer-dimer model on general finite graphs with
bounded degrees. Under the finite fourth moment assumption on the weight
distributions, we prove a Gaussian central limit theorem for the free energy of
the associated Gibbs measure with a rate of convergence. The central limit
theorem continues to hold under a nearly optimal finite $(2+\epsilon)$-moment
assumption on the weight distributions if the underlying graphs are further
assumed to have a uniformly subexponential volume growth. This generalizes a
recent result by Dey and Krishnan (arXiv:2109.12716) who showed a Gaussian
central limit theorem in the disordered monomer-dimer model on cylinder graphs.
Our proof relies on the idea that the disordered monomer-dimer model exhibits a
decay of correlation with high probability. We also establish a central limit
theorem for the Gibbs average of the number of dimers where the underlying
graph has subexponential volume growth and the edge weights are Gaussians.
