Talk by Kiyan Naderi

A metric graph G is a metric space consisting of vertices and intervals connecting the vertices. If we now consider a measure on this space, we obtain a corresponding first-order Sobolev space, with the energy form as its inner product. For every measure we obtain a self-adjoint Laplace operator generated by the energy. We investigate lower bounds for the k-th eigenvalue of all these Laplacians. More precisely, we define the so-called k-th optimal eigenvalue as the infimum of all k-th eigenvalues of Laplace operators generated by Borel measures. First properties of these new graph invariants are investigated: On the one hand, the first optimal eigenvalue is closely related to the resistance metric for graphs, random walks, and spectral graph partitioning. On the other hand, we have studied the asymptotic behavior of the higher optimal eigenvalues and proved a Weyl law.