Algebraic Combinatorics of Magic Squares

2004/05/25 by Maya Mohsin Ahmed, Ahmed, Maya Mohsin · 3 citations
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Graph Labeling and Dimension Problems #graph theory and CDMA systems #math.CO

paper · pdf · doi:10.48550/arxiv.math/0405476

Ph.D. Thesis

arxiv created 2004/05/25 · arxiv updated 2009/12/01

Abstract

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.

Cited by

• Constructing All Magic Squares of Order Three
• Magic graphs and the faces of the Birkhoff polytope
• A generalization of product construction of multimagic squares
• Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex\n Method

Related

• Magic graphs and the faces of the Birkhoff polytope
• Polyhedral Cones of Magic Cubes and Squares
• Enumerating Magic Distinct Labellings of the Cube
• The Martin Gardner Polytopes
• Magic Polygons and Their Properties