Higher matrix-tree theorems and Bernardi polynomial
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The classical matrix-tree theorem discovered by Kirchhoff in 1847 expresses the principal minor of the \(n \times n\) Laplace matrix as a sum of monomials of matrix elements indexed by directed trees with n vertices. We prove, for any \(k \ge n\), a three-parameter family of identities between degree k polynomials of matrix elements of the Laplace matrix. For \(k=n\) and special values of the parameters, the identity turns to be the matrix-tree theorem. For the same values of parameters and arbitrary \(k \ge n\), the left-hand side of the identity becomes a specific polynomial of the matrix elements called higher determinant of the matrix. We study properties of the higher determinants; in particular, they have an application (due to M. Polyak) in the topology of 3-manifolds.
KeywordsMatrix-tree theorem Directed graph Tutte polynomial
Mathematics Subject Classification05C20 05C31
The research was inspired by numerous discussions with prof. Michael Polyak (Haifa Technion, Israel) whom the author wishes to express his most sincere gratitude.
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