The past decade has seen a pronounced growth of interest in differential operators defined on metric graphs, commonly known as quantum graphs. They offer the technical simplicity of one-dimensional objects, while often displaying complex behaviour akin to higher-dimensional ones, and are thus particularly useful as toy models. Starting with a brief introduction to the variational analysis and spectral theory of differential operators on graphs, we will present some recent results on eigenvalue estimates for operators of Laplacian and p-Laplacian type obtainable by variational means.

This is partly based on joint work with Gregory Berkolaiko (Texas A&M), Pavel Kurasov (Stockholm) and Delio Mugnolo (Hagen).