Determinants in the Kronecker product of matrices: the incidence matrix of a complete graph

Abstract

We investigate the least common multiple of all subdeterminants, lcmd(A ⊗ B), of a Kronecker product of matrices, of which one is an integral matrix A with two columns and the other is the incidence matrix of a complete graph with n vertices. We prove that this quantity is the least common multiple of lcmd(A) to the power n − 1 and certain binomial functions of the entries of A.

Keywords:

• Kronecker product
• determinant
• least common multiple
• incidence matrix of complete graph
• matrix minor

Mathematics Subject Classifications::

• 15A15
• 05C50
• 15A57

Acknowledgements

The authors would like to thank an anonymous referee whose suggestions greatly improved the readability of our article.