A generalization of the all minors matrix tree theorem to semirings
This paper extends to semirings the `all minors matrix tree theorem', which is a generalization of the ``matrix tree theorem''. The latter relates the values of the determinant of square submatrices of a given square matrix to the sum of the weights of some families of forests in the associated weighted graph. Since a semiring is no longer assumed to be a group with respect to addition, the combinatorial proof given significantly differs from the one working on the fields of real numbers. Even the statement of the result has to be changed, since the standard concept of determinant of a matrix is not relevant any more and must be replaced by the concept of bideterminant.
scientific article; zbMATH DE number 1321797