graph with $n$ vertices and $e$ edges has the fewest matchings. (A *matching* in a graph is simply a set of pairwise non-incident edges.) We denote the set of matchings in $G$ by ${\mathcal{M}}(G)$ and the set of $k$-matchings in $G$ by ${\mathcal{M}}_k(G)$. Also we write $m(G)=|{\mathcal{M}}(G)|$ and $m_k(G)= |{\mathcal{M}}_k(G)|$. In general our notation is standard and follows that of Bollobás [@MGT]. It turns out that, following the general approach of Ahlswede and Katona, we need to consider the class ${\mathcal{B}}_{\ell,r,e}$ of all bipartite graphs with $\ell$ vertices in the left part, $r$ vertices on the right, and having $e$ edges. We determine the graph in ${\mathcal{B}}_{\ell,r,e}$ having the fewest matchings. Our techniques also allow us to determine the graphs minimizing the number of matchings of size $k$, for all values of $k$, in ${\mathcal{B}}_{\ell,r,e}$ and in ${\mathcal{G}}_{n,e}$. In order to state our results we need to describe three collections of graphs: lex graphs, colex graphs, and lex bipartite graphs.

The *lexicographic order*, $<_L$, on finite subsets of ${\mathbb{N}}$ is defined by $A<_L B$ if $\min(A\Delta B)\in A$. The *colexigraphic order*, $<_C$, is defined by $A<_C B$ if $\max(A\Delta B) \in B$.

Restricting these orderings to $2$-subsets of $[n]$ results 