Higher matrix-tree theorems and Bernardi polynomial
- 18 Downloads
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.
- 1.Awan, J., Bernardi, O.: Tutte polynomials for directed graphs. arXiv:1610.01839v2
- 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
- 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
- 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
- 10.Walker, K.: An Extension of Casson’s Invariant, Annals of Mathematics Studies, 126. Princeton University Press, Princeton (1992)Google Scholar