The role of expanders in the spectral geometry of metric graphs

Expanders are families of graphs that are sparse in edges but dense in connectivity. After reviewing combinatorial and spectral definitions of expanders, we use precise lower bounds on Ramanujan graphs to investigate upper bounds – and, specifically, the lack thereof – on the eigenvalues of the Laplacian on \emph{metric} graphs in terms of volume, diameter, girth, mean distance, and torsional rigidity, among others.