We describe how to construct and enumerate Magic squares, Franklin squares,
Magic cubes, and Magic graphs as lattice points inside polyhedral cones using
techniques from Algebraic Combinatorics. The main tools of our methods are the
Hilbert Poincare series to enumerate lattice points and the Hilbert bases to
generate lattice points. We define polytopes of magic labelings of graphs and
digraphs, and give a description of the faces of the Birkhoff polytope as
polytopes of magic labelings of digraphs.Comment: Ph.D. Thesi
