A generalization of the all minors tree theorem to semirings

Abstract

The ‘All Minors Matrix Tree Theorem’ (Chen, Applied Graph Theory, Graphs and Electrical Networks, North-Holland, Amsterdam, 1976; Chaiken, SIAM J. Algebraic Discrete Math. 3 (3) (1982) 319–329) is an extension of the well-known ‘Matrix Tree Theorem’ (Tutte, Proc. Cambridge Philos. Soc. 44 (1948) 463–482) which relates the values of the determinants of square submatrices of a given square matrix to the sum of the weights of some families of forests in the associated weighted graph. The ‘All Minors Matrix Tree Theorem’ is extended here to algebraic structures much more general than the field of real numbers, namely semirings. Since a semiring is no longer assumed to be a group with respect to the first law (addition), the combinatorial proof given here significantly differs from the one working on the field 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.The proof also significantly differs from the one used in a previous paper to derive the semiring extension of the Matrix Tree Theorem.