  The following problem, which stems from the ``flux phase'' problem in
condensed matter physics, is analyzed and extended here: One is given a planar
graph (or lattice) with prescribed vertices, edges and a weight $\vert
t_{xy}\vert$ on each edge $(x,y)$. The flux phase problem (which we partially
solve) is to find the real phase function on the edges, $\theta(x,y)$, so that
the matrix $T:=\{\vert t_{xy}\vert {\rm exp}[i\theta(x,y)]\}$ minimizes the sum
of the negative eigenvalues of $-T$. One extension of this problem which is
also partially solved is the analogous question for the Falicov-Kimball model.
There one replaces the matrix $-T$ by $-T+V$, where $V$ is a diagonal matrix
representing a potential. Another extension of this problem, which we solve
completely for planar, bipartite graphs, is to maximize $\vert {\rm det}\ T
\vert$. Our analysis of this determinant problem is closely connected with
Kasteleyn's 1961 theorem (for arbitrary planar graphs) and, indeed, yields an
alternate, and we believe more transparent proof of it. {}.
