A generalization of the all minors matrix tree theorem to semirings
type: Item
instance of: scholarly article
title: A generalization of the all minors matrix tree theorem to semirings
zbMATH Open document ID: 0928.15010
DOI: 10.1016/S0012-365X(98)00204-0
published in: Discrete Mathematics
publication date: 2000-01-09T00:00:00Z
review text: 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.
reviewed by: Giacomo Bonanno
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 1321797
zbMATH Keywords: bideterminant
zbMATH Keywords: forest
zbMATH Keywords: semirings
zbMATH Keywords: minors
zbMATH Keywords: tree
zbMATH Keywords: determinant
zbMATH Keywords: weighted graph
author: Michel Minoux
MaRDI profile type: Publication
cites work: A Combinatorial Proof of the All Minors Matrix Tree Theorem
cites work: Q4088840
cites work: Q3684177
cites work: Line-digraphs, arborescences and theorems of Tutte and Knuth
cites work: Q5786986
full work available at URL: s0012-365x(98)00204-0
OpenAlex ID: W2060388682
Wikidata QID: Q126383022
description: scientific article; zbMATH DE number 1321797