  In recent years there has been increased interest in extremal problems for
"counting" parameters of graphs. For example, the Kahn-Zhao theorem gives an
upper bound on the number of independent sets in a $d$-regular graph. In the
same spirit, the Upper Matching Conjecture claims an upper bound on the number
of $k$-matchings in a $d$-regular graph. Here we consider both matchings and
matchings of fixed sizes in graphs with a given number vertices and edges. We
prove that the graph with the fewest matchings is either the lex or the colex
graph. Similarly, for fixed $k$, the graph with the fewest $k$-matchings is
either the lex or the colex graph. To prove these results we first prove that
the lex bipartite graph has the fewest matchings of all sizes among bipartite
graphs with fixed part sizes and a given number of edges.
