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Manifestly non-Hermitian quantum graphs with real spectra are introduced andshown tractable as a new class of phenomenological models with severalappealing descriptive properties. For illustrative purposes, just equilateralstar-graphs are considered here in detail, with non-Hermiticities introduced byinteractions attached to the vertices. The facilitated feasibility of theanalysis of their spectra is achieved via their systematic approximativeRunge-Kutta-inspired reduction to star-shaped discrete lattices. The resultingbound-state spectra are found real in a discretization-independent interval ofcouplings. This conclusion is reinterpreted as the existence of a hiddenHermiticity of our models, i.e., as the standard and manifest Hermiticity ofthe underlying Hamiltonian in one of less usual, {\em ad hoc} representations${\cal H}_j$ of the Hilbert space of states in which the inner product is local(at $j=0$) or increasingly nonlocal (at $j=1,2, ...$). Explicit examples ofthese (of course, Hamiltonian-dependent) hermitizing inner products are offeredin closed form. In this way each initial quantum graph is assigned a menu ofoptional, non-equivalent standard probabilistic interpretations exhibiting acontrolled, tunable nonlocality.

Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs