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

TL;DR

This paper explores the least common multiple of subdeterminants in a Kronecker product involving an integral matrix and a complete graph's incidence matrix, revealing a formula linking it to the original matrix and combinatorial functions.

Contribution

It provides a novel formula for the least common multiple of subdeterminants in a specific Kronecker product involving incidence matrices.

Findings

lcmd(A x B) equals lcmd(A)^{n-1} times binomial functions of A's entries

The result connects matrix determinants with combinatorial properties of complete graphs

Theoretical insight into subdeterminants of Kronecker products involving incidence matrices

Abstract

We investigate the least common multiple of all subdeterminants, lcmd(A x 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.