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We study the spectra of quantum graphs with the method of trace identities(sum rules), which are used to derive inequalities of Lieb-Thirring,Payne-P\'olya-Weinberger, and Yang types, among others. We show that the sharpconstants of these inequalities and even their forms depend on the topology ofthe graph. Conditions are identified under which the sharp constants are thesame as for the classical inequalities; in particular, this is true in the caseof trees. We also provide some counterexamples where the classical form of theinequalities is false.

On semiclassical and universal inequalities for eigenvalues of quantum graphs