\Gamma_{\beta} := \bigcup_{j=0}^{N-1}
L_j$ to which we associate[^7] the Hamiltonian $H_N(\beta):=H_{\alpha,\Gamma_\beta}$. Trivial examples are

1.  $\,H_2(\pi)\,$ corresponding to a straight line which has obviously a purely a.c. spectrum covering the interval $[-\frac14
    \alpha^2, \infty)$,

2.  $\,H_4(\beta_s)\,$ with $\beta_s= \left\lbrace {\pi\over 2},
    {\pi\over 2}, {\pi\over 2} \right\rbrace$ corresponding to cross-shaped $\Gamma$ allows again a separation of variables. The a.c. part of its spectrum is the same as above, and in addition, there is a single isolated eigenvalue $-\frac12\alpha^2$ corresponding to the eigenfunction $(2\alpha)^{-1}
    {\mathrm{e}}^{-\alpha(|x|+|y|)/2}$.

Star-shaped graphs have the property indicated above:

\[star\_ess\] $\,\sigma_\mathrm{ess}(H_N(\beta)) = [-\frac14 \alpha^2, \infty)$ holds for any $N$ and $\beta$.

The inequality $\,\inf \sigma_\mathrm{ess}(H_N(\beta)) \ge
-\frac14 \alpha^2\,$ is obtained by Neumann bracketing dissecting the plane outside a compact into semi-infinite strips with $L_j$ in the middle and “empty” wedges. The fact that $[-\frac14
\alpha^2, \infty)$ belongs to the spectrum is checked by means of a family of Weyl sequences, cf. [@EN1].

By Corollary \[dsgraph\], $\:\sigma_\mathrm{disc}
(H_N(\beta))\,$ is nonempty unless $N=2$ and $\beta=\pi$. Using direct methods one can establish various other properties of the discrete spectrum[^8].

\[star\_many\_bs\] Fix $N$ and a positive integer $n$. If at least one of the angles $\beta_j$ is small enough, $\sharp\, \sigma_\mathrm{disc}
(H_N(\beta)) \ge 