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Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes. Magic cubes and squares are very popular combinatorial objects (see [2, 15, 17] and their references). A magic square is a square matrix whose entries are nonnegative… (More)

- Maya Mohsin Ahmed
- The American Mathematical Monthly
- 2004

In this article, we construct and enumerate the magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. We explore further the correspondence of magic labelings of the complete general graph and symmetric magic squares. We define magic digraphs and establish a correspondence between semi-magic squares… (More)

- M M Ahmed
- Acta anatomica
- 1970

- M Ahmed, J A Wimhurst, N P Walton
- Injury
- 2007

Magic labelings of graphs are studied in great detail by Stanley in [18] and [19], and Stewart in [20] and [21]. In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. We define polytopes of magic labelings of graphs and digraphs. We give a description of the… (More)

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