Coadjoint pair

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Coadjoint pairсопряженная пара.

A pair of operators \,(A,P) is a coadjoint pair if \,A is an adjacency operator \,A(G) for a graph \,G and P = \sum_{v \in V(G)} \varphi(v) \otimes v is a permutation on \,V(G) satisfying

A(G)^{\ast} = P^{\ast}A(G)P.

Moreover, the bijection \varphi on \,V(G) satisfies \varphi^{2}
=1, or \,P^{2} = 1. In this case, \,P is called a transposition symmetry. Like this case, if a graph \,G has a coadjoint pair \,(A,S) such that \,S is a transposition symmetry, then \,G is called strongly coadjoint. Needless to say, undirected graphs are all strongly coadjoint and strongly coadjoint graphs are all coadjoint.


  • Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.