subset of ${\mathbb{R}}^2$ contains at most a finite number of edges and vertices of $\Gamma$.

The last assumption allows us to extend Theorem 4.1 of [@BEKS] applying it to the Dirac measure supported by the graph. More exactly, we consider the measure $$\label{meas}
m_\Gamma:\; m_\Gamma(M) = \ell_1(M\cap\Gamma)$$ for any Borel $M\subset{\mathbb{R}}^2$, where $\ell_1$ is the one-dimensional Hausdorff measure given in our case by the edge-arc length. Such a straightforward extension implies that $m\equiv m_\Gamma$ satisfies the condition (\[basiccond\]) and gives thus rise to the appropriate operator $\hat H_{-\alpha m}$; to make the notation explicit we will employ for it in the following the symbol $H_{\alpha,\Gamma}$. This is one way how to give meaning to the formal expression (\[formal\]).

An alternative is to use boundary conditions. Consider the operator acting as $$\left(\dot H_{\alpha, \Gamma}\psi \right)(x) =
-(\Delta\psi)(x)\,, \quad x\in {\mathbb{R}}^2\setminus\Gamma\,,$$ on any function $\psi$ which belongs to $W^{2,2}({\mathbb{R}}^2\setminus\Gamma)$, is continuous at each edge $e_j\in\Gamma$ with the normal derivatives having there a jump, namely $$\label{2-1bc}
 {\partial\psi\over\partial n_+}(x) - {\partial\psi\over\partial
n_-}(x) = -\alpha \psi(x)\,, \quad x\in \mathrm{int\,}e_j\,;$$ since the edges are smooth by assumption, the normal vector exists at each inner point of an edge. In the same way as in [@BEKS] one 