It may seem that the result covers only a rather particular class of graphs. Using the minimax principle, however, we arrive at the following easy consequence:

\[dsgraph\] Suppose that $\Gamma$ has a subgraph in the form of an infinite curve satisfying the assumptions of the theorem, and $\sigma_\mathrm{ess}(H_{\alpha,\Gamma}) = \left\lbrack -\frac14
\alpha^2, \infty \right)$, then the discrete spectrum of $H_{\alpha,\Gamma}$ is non-empty.

It is important that the assumption about preservation of the essential spectrum can be often verified easily, for instance, in the situation when $\Gamma$ has outside a compact a finite number of straight edges separated by non-trivial wedges.

An example: leaky star graphs {#ss: star}
-----------------------------

To illustrate the last made claim, let us investigate in more detail a particular class of such graphs, namely the situation when $\Gamma$ is of a *star shape*. Given an integer $N\ge
2$, consider an $(N\!-\!1)$-tuple $\beta=\{ \beta_1,\dots,
\beta_{N-1}\}$ of positive numbers such that $$\beta_N := 2\pi - \sum_{j=1}^{N-1} \beta_j > 0\,.$$ Denote $\vartheta_j:= \sum_{i=1}^j \beta_j$ and $\vartheta_0:=0$. Let $L_j$ be the radial half-line with the endpoint at the origin, $L_j:= \{ x\in{\mathbb{R}}^2:\; \arg x=\vartheta_j \}$, naturally parametrized by its arc length $s=|x|$. These half-lines will be the edges of $\Gamma \equiv 