  The Faber-Krahn inequality states that among all open domains with a fixed
volume in R^n, the ball minimizes the first Dirichlet eigenvalue of the
Laplacian. We study an asymptotic discrete analogue of this for the
combinatorial Dirichlet Laplacian acting on induced subgraphs of Z^2. Namely,
an induced subgraph G with n vertices is called a minimizing subgraph if it
minimizes the first eigenvalue of the combinatorial Dirichlet Laplacian among
all induced subgraphs with n vertices. Consider an induced subgraph G and take
the interior of the union of closed squares of area 1 about each point of G.
Let G* denote this domain scaled down to have area 1. Our main theorem states
that if {G_n} is a sequence of minimizing subgraphs where each G_n has n
vertices, then after translation the measure of the symmetric difference of
G_n* and the unit disk converges to 0.
