Higher matrix-tree theorems and Bernardi polynomial

Abstract

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.

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Acknowledgements

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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Correspondence to Yurii Burman.

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The research was funded by the Russian Academic Excellence Project ‘5-100’ and by the Simons-IUM fellowship 2017 by the Simons Foundation.

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Burman, Y. Higher matrix-tree theorems and Bernardi polynomial. J Algebr Comb 50, 427–446 (2019). https://doi.org/10.1007/s10801-018-0863-x

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Keywords

  • Matrix-tree theorem
  • Directed graph
  • Tutte polynomial

Mathematics Subject Classification

  • 05C20
  • 05C31