Investigations of Graph Polynomials
نویسندگان
چکیده
Investigations of graph polynomials Mirkó Visontai Advisor: James Haglund This thesis consists of two parts. The first part is a brief introduction to graph polynomials. We define the matching, rook and hit polynomials, reveal the connection between them and show necessary conditions that imply that all roots of these polynomials are real. In the second part, we focus on the closely related Monotone Column Permanent (MCP) conjecture of Haglund, Ono, and Wagner [HOW99]. This conjecture is known to be true for n ≤ 3 [Hag00] and is open for n ≥ 4. We present our following new results on this conjecture. First, we give an alternative proof for the n = 2 case. Then, we generalize the MCP conjecture for non-square matrices. Further developing the idea of the proof used in the n = 2 case, we show that the conjecture holds for matrices of size n× 2 and 2×m as well. Finally, we investigate a different approach in attempt to prove the conjecture. We obtain partial results by proving that the MCP conjecture for n × n matrices implies the MCP conjecture for n ×m matrices for n ≤ m. We also show a conditional result that if the MCP conjecture is true for n×n then it is also true for the (n+ 1)× (n+ 1) case under the assumption that the permanents of certain minors of the matrix have interlacing roots.
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تاریخ انتشار 2007