##
**Determinants in the Kronecker product of matrices: the incidence matrix of a complete graph.**
*(English)*
Zbl 1218.15003

Summary: We investigate the least common multiple of all subdeterminants, lcmd\((A \otimes B)\), 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 \(A\).

### MSC:

15A15 | Determinants, permanents, traces, other special matrix functions |

11A05 | Multiplicative structure; Euclidean algorithm; greatest common divisors |

05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |

15B36 | Matrices of integers |

15A69 | Multilinear algebra, tensor calculus |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

### Keywords:

least common multiple; subdeterminants; Kronecker product; integral matrix; incidence matrix; complete graph### References:

[1] | Chaiken S, A q-queens problem |

[2] | Horn RA, Topics in Matrix Analysis (1991) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.