  Toy quantum Hamiltonians $H\neq H^\dagger$ with real spectra are considered
as living on graphs $\mathbb{G}$ which only differ from the standard real line
$\mathbb{R}$ locally, on a microscopic fundamental-length scale. In terms of a
nontrivial metric the "hidden Hermiticity" property $H= H^\ddagger$ is
postulated. Our calculations of the energies (based on a discretization of
$\mathbb{G}$) indicate that the nontriviality of the topology of the graph may
be responsible for certain nonperturbative features of the energies.
