Algebraic Combinatorics of Magic Squares

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

Application of the Combinatorial Nullstellensatz to Integer-magic Graph Labelings

Let $A$ be a nontrivial abelian group and $A^* = A \setminus \{0\}$. A graph is $A$-magic if there exists an edge labeling $f$ using elements of $A^*$ which induces a constant vertex labeling of the graph. Such a labeling $f$ is called an $A$-magic labeling and the constant value of the induced vertex labeling is called an $A$-magic value. In this paper, we use the Combinatorial Nullstellensatz to show the existence of $\mathbb{Z}_p$-magic labelings (prime $p \geq 3$ ) for various graphs, without having to construct the $\mathbb{Z}_p$-magic labelings. Through many examples, we illustrate the usefulness (and limitations) in applying the Combinatorial Nullstellensatz to the integer-magic labeling problem. Finally, we focus on $\mathbb{Z}_3$-magic labelings and give some results for various classes of graphs

Finite type invariants of cyclic branched covers

Updated rerefences and introduction. Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its signature function. Revised version.Comment: LaTeX, 28 pages with 22 figure

Inside-Out Polytopes

We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat