Georgia Institute of Technology College of Sciences

I'll review a few known results about quantum graphs that maximize or minimize eigenvalues or combinations of eigenvalues, and will then concentrate on ratios of eigevalues under topological constraints on the graph.  In particular a new discovery with James Kennedy and Gabriel Ramos is that the largest ratio of the first two eigenvalues of the Laplacian on a finite tree graph with Dirichlet conditions at the ends is achieved by equilateral stars. Some related, Weyl-sharp estimates of arbitrary eigenvalue ratios can be obtained using similar ideas.  If time permits, I will also describe some optimal results about differences and other combinations of eigenvalues.