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The paper deals with some spectral properties of (mostly infinite) quantumand combinatorial graphs. Quantum graphs have been intensively studied latelydue to their numerous applications to mesoscopic physics, nanotechnology,optics, and other areas.  A Schnol type theorem is proven that allows one to detect that a pointbelongs to the spectrum when a generalized eigenfunction with an subexponentialgrowth integral estimate is available. A theorem on spectral gap opening for``decorated'' quantum graphs is established (its analog is known for thecombinatorial case). It is also shown that if a periodic combinatorial orquantum graph has a point spectrum, it is generated by compactly supportedeigenfunctions (``scars'').Comment: 4 eps figures, LATEX file, 21 pages Revised form: a cut-and-paste  blooper fixe

Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs