  Although the physical Hamiltonian operator can be constructed in the
deparameterized model of loop quantum gravity coupled to a scalar field, its
property is still unknown. This open issue is attacked in this paper by
considering an operator $\hat{H}_v$ representing the square of the physical
Hamiltonian operator acting nontrivially on two-valent spin networks. The
Hilbert space $\mathcal{H}_v$ preserved by the graphing changing operator
$\hat{H}_v$ is consist of spin networks with a single two-valent non-degenerate
vertex. The matrix element of $\hat{H}_v$ are explicitly worked out in a
suitable basis. It turns out that the operator $\hat{H}_v$ is essentially
self-adjoint, which implies a well-defined physical Hamiltonian operator in
$\mathcal{H}_v$ for the deparameterized model.
