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We consider Schr\"odinger operators on a class of periodic quantum graphswith randomly distributed Kirchhoff coupling constants at all vertices. Usingthe technique of self-adjoint extensions we obtain conditions for localizationon quantum graphs in terms of finite volume criteria for some energy-dependentdiscrete Hamiltonians. These conditions hold in the strong disorder limit andat the spectral edges.

Localization on quantum graphs with random vertex couplings