The Graham-Pollack Theorem and Its Extensions

Title: The Graham-Pollack Theorem and Its Extensions

Research Question: How can we extend the Graham-Pollack theorem to different fields and contexts, and what are the implications of these extensions?

Methodology: The researchers used algebraic reasoning and combinatorial methods to extend the Graham-Pollack theorem to different fields and contexts. They considered problems related to covering a complete graph by complete bipartite graphs an odd number of times and depth-3 arithmetic circuits for S2 n(X).

Results: The researchers found that for infinitely many n, the minimum number of complete bipartite graphs required to cover each edge of a complete graph an odd number of times is /ceilingleftbign 2/ceilingrightbig /. They also determined exactly the number of multiplication gates required for depth-3 arithmetic circuits of the form /summationtext r for inhomogeneous circuits where the Lij’s are allowed to have constants, and for homogeneous models.

Implications: These extensions of the Graham-Pollack theorem have implications for various areas of mathematics and computer science. They provide insights into the complexity of covering problems and the design of arithmetic circuits, which can be applied in areas such as cryptography, computer graphics, and artificial intelligence.

Conclusion: The researchers have extended the Graham-Pollack theorem to different fields and contexts, providing new insights into the complexity of covering problems and the design of arithmetic circuits. These extensions have implications for various areas of mathematics and computer science, making this research valuable for the broader scientific community.

Link to Article: https://arxiv.org/abs/0110031v1 Authors: arXiv ID: 0110031v1