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On the separation of eigenvalues by the permutation group

100%
Cigler G., Jerman M.
Special Matrices
|
2014
|
tom 2
|
nr 1
61-67
EN
Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.
2
Content available remote

On separation of eigenvalues by the permutation group

100%
Cigler G., Jerman M.
Special Matrices
|
2014
|
tom 2
|
nr 1
78-84
EN
Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.
3
Content available remote

Chromatic Properties of the Pancake Graphs

88%
Konstantinova E.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
777-787
EN
Chromatic properties of the Pancake graphs Pn, n ⩾ 2, that are Cayley graphs on the symmetric group Symn generated by prefix-reversals are investigated in the paper. It is proved that for any n ⩾ 3 the total chromatic number of Pn is n, and it is shown that the chromatic index of Pn is n − 1. We present upper bounds on the chromatic number of the Pancake graphs Pn, which improve Brooks’ bound for n ⩾ 7 and Katlin’s bound for n ⩽ 28. Algorithms of a total n-coloring and a proper (n − 1)-coloring are given.
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