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On Polynomials of Spanning Trees
Fan Chung1, Chao Yang2
1University of California,San Diego, La Jolla, CA92093-0112, USA
fan@euclid.ucsd.edu
2University of Pennsylvania, Philadelphia, PA 19104, USA
Annals of Combinatorics 4 (1) p.13-25 March, 2000
AMS Subject Classification:05E99
Abstract:
The Kirchhoff polynomial of a graph G is the sum of weights of all spanning trees where the weight of a tree is the product of all its edge weights, considered as formal variables. Kontsevich conjectured that when edge weights are assigned values in a finite field $ {\mathbb{F}}_q$, for a prime power q, the number of zeros of the Kirchhoff polynomial of a graph G is just a polynomial function of q. Stanley verified this conjecture for some families of graphs and further proposed several conjectures. In this paper, we derive and prove explicit formulas for certain graphs, thus confirming Stanley's conjectures. We also consider several extensions and generalizations of the conjecture of Kontsevich.
Keywords: Kirchhoff polynomials, matrix-tree theorem, probabilistic Kirchhoff polynomial, combinatorial Laplacian, spanning trees

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