  In this paper we review the asymptotic matching conjectures for $r$-regular
bipartite graphs, and their connections in estimating the monomer-dimer
entropies in $d$-dimensional integer lattice and Bethe lattices. We prove new
rigorous upper and lower bounds for the monomer-dimer entropies, which support
these conjectures. We describe a general construction of infinite families of
$r$-regular tori graphs and give algorithms for computing the monomer-dimer
entropy of density $p$, for any $p\in [0,1]$, for these graphs. Finally we use
tori graphs to test the asymptotic matching conjectures for certain infinite
$r$-regular bipartite graphs.
