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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.