Spectral Inequalities for Quantum Graphs

We review our joint work with Evans Harrell on semiclassical and universal inequalities for quantum graphs. The proofs of these inequalities are based on an abstract trace inequality for commutators of operators. In this article we give a new proof of this abstract trace inequality. Another ingredient in proving semiclassical and universal inequalities is an appropriate choice of operators in this trace inequality. We provide a new approximation method for such a choice