Coadjoint pair

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.