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Higher matrix-tree theorems and Bernardi polynomial

  • Yurii BurmanEmail author
Article
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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.

Keywords

Matrix-tree theorem Directed graph Tutte polynomial 

Mathematics Subject Classification

05C20 05C31 

Notes

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.

References

  1. 1.
    Awan, J., Bernardi, O.: Tutte polynomials for directed graphs. arXiv:1610.01839v2
  2. 2.
    Burman, Yu., Ploskonosov, A., Trofimova, A.: Matrix-tree theorems and discrete path integration. Linear Algebra Appl. 466, 64–82 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chaiken, S.: A combinatorial proof of the all minors matrix tree theorem. SIAM J. Discrete Math. 3(3), 319–329 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Epstein, B.: A combinatorial invariant of \(3\)-manifolds via cycle-rooted trees. MSc. thesis (under supervision of prof. M. Polyak), Technion, Haifa, Israel (2015)Google Scholar
  5. 5.
    Kirchhoff, G.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme gefurht wird. Ann. Phys. Chem. 72, 497–508 (1847)CrossRefGoogle Scholar
  6. 6.
    Polyak, M.: From \(3\)-manifolds to planar graphs and cycle-rooted trees, talk at Arnold’s legacy conference. Fields Institute, Toronto (2014)Google Scholar
  7. 7.
    Rota, G.-C.: On the foundations of combinatorial theory I: theory of Möbius functions. Z. Wahrsch. Verw. Gebiete 2, 340–368 (1964)CrossRefGoogle Scholar
  8. 8.
    Sokal, A.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Webb, B.S. (ed.) Surveys in combinatorics 2005. London Mathematical Society Lecture Note Series, vol. 327, pp. 173–226. Cambridge University Press, Cambridge (2005)Google Scholar
  9. 9.
    Tutte, W.T.: The dissection of equilateral triangles into equilateral triangles. Math. Proc. Cambridge Philos. Soc. 44(4), 463–482 (1948)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Walker, K.: An Extension of Casson’s Invariant, Annals of Mathematics Studies, 126. Princeton University Press, Princeton (1992)Google Scholar
  11. 11.
    Welsh, D.J.A., Merino, C.: The Potts model and the Tutte polynomial. J. Math. Phys. 41(3), 1127–1152 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.Independent University of MoscowMoscowRussia

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