Higher matrix-tree theorems and Bernardi polynomial

Journal of Algebraic Combinatorics. 2019. Vol. 50. No. 4. P. 427–446.

Burman Y. M.

The classical matrix-tree theorem discovered by G.Kirchhoff in 1847
expresses the principal minor of the (n x n) Laplace matrix as a
sum of monomials of matrix elements indexed by directed trees with n
vertices. We prove, for any k >= 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 >= 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: определитель, determinant, matrix-tree theorem, Tutte polynomial, Матричная теорема о деревьях, многочлен Татта

Publication based on the results of: Теория представлений и математическая физика (2018)