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We introduce a dynamically defined class of unbounded, connected, equilateralmetric graphs on which the Kirchhoff Laplacian has zero Lebesgue measurespectrum and a nontrivial singular continuous part. A new local Borg--Marchenkouniqueness result is obtained in order to utilize Kotani theory for aperiodicsubshifts satisfying Boshernitzan's condition.

Zero Measure and Singular Continuous Spectra for Quantum Graphs