|\nabla\psi(x)|^2 \, {\mathrm{d}}x
-\alpha \int_{{\mathbb{R}}^2} |(I_m\psi)(x)|^2\, {\mathrm{d}}m(x)$$ is defined on $W^{1,2}({\mathbb{R}}^2)$; it is straightforward to check [@BEKS] that under the condition (\[basiccond\]) this form is closed and below bounded, with $C_0^{\infty}({\mathbb{R}}^2)$ as a core, and consequently, it is associated with a unique self-adjoint operator denoted as $\hat H_{-\alpha m}$. A sufficient condition for the inequality (\[basiccond\]) to be valid is that the measure $m$ belongs to the generalized Kato class, i.e. $$\label{Kato}
\lim_{\epsilon\to 0}\: \sup_{x\in{\mathbb{R}}^2}\, \int_{B_\epsilon(x)} |\ln
|x\!-\!y|| \, {\mathrm{d}}m(y) = 0\,,$$ where $B_\epsilon(x)$ is the ball of radius $\epsilon$ centered at $x$. In such a case, moreover, any positive number can be chosen as $a$. So far the construction has been general and involved also regular Schrödinger operators. Suppose now that $m$ is the Dirac measure supported by a graph $\Gamma\subset{\mathbb{R}}^2$ which has the following properties:

*(g1)* edge smoothness

:   each edge $e_j\in\Gamma$ is a graph of $C^1$ function $\gamma_j:\:I_j\to{\mathbb{R}}^2$ where $I_j$ is an interval (finite, semi-infinite, or the whole ${\mathbb{R}}$). Moreover, without loss of generality we may suppose that edges are parametrized by the arc length, $|\dot\gamma_j(s)|=1$.

*(g2)* cusp absence

:   at the vertices of $\Gamma$ the edges meet at nonzero angles.

*(g3)* local finiteness

:   each compact 