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A directed infinite graph $\,G$ is a pair of the set $\,V$ of the countable vertices and the set $\,E$ of the arrows (arcs) $u \leftarrow v$. Let ${\mathcal H}$ be a Hilbert space $\ell^{2}(G)$ on $\,V$ with a canonical basis $\{e_{v}| \; v \in V\}$. Since every arrow $u \leftarrow v \in E$ induces a dyad $e_{u} \otimes e_{v}$, where $(x \otimes y)z = \langle z,y\rangle x$ for $x, y, z \in {\mathcal H}$, the adjacency operator $\,A(G)$ is expressed by

$A(G) = \sum_{u \leftarrow v} e_{u} \otimes e_{v}$

if $\,G$ has a bounded degree.

Adjacency operators are classified as follows:

• $\,A$ is self-adjoint if $A = A^{\ast}$.
• $\,A$ is unitary if $A^{\ast}A = AA^{\ast} = I$.
• $\,A$ is normal if $A^{\ast}A = AA^{\ast}$.
• $\,A$ is hyponormal (resp. co-hyponormal) if $A^{\ast}A \geq AA^{\ast}$ (resp. $AA^{\ast} \geq A^{\ast}A$).
• $\,A$ is projection if $A = A^{\ast} = A^{2}$.
• $\,A$ is partial isometry if $A^{\ast}A$ and $AA^{\ast}$.
• $\,A$ is isometry (resp. co-isometry) if $A^{\ast}A = I$ (resp.$AA^{\ast} = I$).
• $\,A$ is nilpotent if there exists a number $\,n$ such that $\,A^{n} = 0$.
• $\,A$ is idempotent if $\,A = A^{2}$.
• $\,A$ is positive if $(Ax|x) \geq 0$ for $x \in H$.

Here $G^{\ast}$ is the adjoint graph for a graph $\,G$.