Coadjoint pair: различия между версиями
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'''Coadjoint pair''' | '''Coadjoint pair''' — ''[[сопряженная пара]].'' | ||
A pair of operators <math>(A,P)</math> is a '''coadjoint pair''' if <math>A</math> is an ''adjacency operator'' <math>A(G)</math> for a graph <math>G</math> and <math>P = \sum_{v \in V(G)} | A pair of operators <math>\,(A,P)</math> is a '''coadjoint pair''' if <math>\,A</math> is an ''[[adjacency operator]]'' <math>\,A(G)</math> for a [[graph, undirected graph, nonoriented graph|graph]] <math>\,G</math> and <math>P = \sum_{v \in V(G)} \varphi(v) \otimes v</math> is a permutation on <math>\,V(G)</math> satisfying | ||
\varphi(v) \otimes v</math> is a permutation on <math>V(G)</math> satisfying | |||
<math>A(G)^{\ast} = P^{\ast}A(G)P.</math> | ::::::<math>A(G)^{\ast} = P^{\ast}A(G)P.</math> | ||
Moreover, the bijection <math>\varphi</math> on <math>V(G)</math> satisfies <math>\varphi^{2} | Moreover, the bijection <math>\varphi</math> on <math>\,V(G)</math> satisfies <math>\varphi^{2} | ||
=1</math>, or <math>P^{2} = 1</math>. In this case, <math>P</math> is called a '''transposition symmetry'''. Like this case, if a graph <math>G</math> has a '''coadjoint pair''' <math>(A,S)</math> | =1</math>, or <math>\,P^{2} = 1</math>. In this case, <math>\,P</math> is called a '''transposition symmetry'''. Like this case, if a graph <math>\,G</math> has a '''coadjoint pair''' <math>\,(A,S)</math> such that <math>\,S</math> is a transposition symmetry, then <math>\,G</math> is called '''strongly coadjoint'''. Needless to say, undirected graphs are all | ||
such that <math>S</math> is a transposition symmetry, then <math>G</math> is called '''strongly coadjoint'''. Needless to say, undirected graphs are all | |||
strongly coadjoint and strongly coadjoint graphs are all coadjoint. | strongly coadjoint and strongly coadjoint graphs are all coadjoint. | ||
==Литература== | |||
* Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009. | |||
Текущая версия от 10:47, 24 октября 2018
Coadjoint pair — сопряженная пара.
A pair of operators [math]\displaystyle{ \,(A,P) }[/math] is a coadjoint pair if [math]\displaystyle{ \,A }[/math] is an adjacency operator [math]\displaystyle{ \,A(G) }[/math] for a graph [math]\displaystyle{ \,G }[/math] and [math]\displaystyle{ P = \sum_{v \in V(G)} \varphi(v) \otimes v }[/math] is a permutation on [math]\displaystyle{ \,V(G) }[/math] satisfying
- [math]\displaystyle{ A(G)^{\ast} = P^{\ast}A(G)P. }[/math]
Moreover, the bijection [math]\displaystyle{ \varphi }[/math] on [math]\displaystyle{ \,V(G) }[/math] satisfies [math]\displaystyle{ \varphi^{2} =1 }[/math], or [math]\displaystyle{ \,P^{2} = 1 }[/math]. In this case, [math]\displaystyle{ \,P }[/math] is called a transposition symmetry. Like this case, if a graph [math]\displaystyle{ \,G }[/math] has a coadjoint pair [math]\displaystyle{ \,(A,S) }[/math] such that [math]\displaystyle{ \,S }[/math] is a transposition symmetry, then [math]\displaystyle{ \,G }[/math] is called strongly coadjoint. Needless to say, undirected graphs are all strongly coadjoint and strongly coadjoint graphs are all coadjoint.
Литература
- Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.