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We consider discrete Schr\"odinger operators with periodic potentials onperiodic graphs perturbed by guided non-positive potentials, which are periodicin some directions and finitely supported in other ones. The spectrum of theunperturbed operator is a union of a finite number of non-degenerate bands andeigenvalues of infinite multiplicity. We show that the spectrum of theperturbed operator consists of the unperturbed one plus the additional guidedspectrum, which is a union of a finite number of bands. We estimate theposition of the guided bands and their length in terms of graph geometricparameters. We also determine the asymptotics of the guided bands for largeguided potentials. Moreover, we show that the possible number of the guidedbands, their length and position can be rather arbitrary for some specificpotentials.

Schr\"odinger operators with guided potentials on periodic graphs