  We derive several upper bounds on the spectral gap of the Laplacian with
standard or Dirichlet vertex conditions on compact metric graphs. In
particular, we obtain estimates based on the length of a shortest cycle
(girth), diameter, total length of the graph, as well as further metric
quantities introduced here for the first time, such as the avoidance diameter.
Using known results about Ramanujan graphs, a class of expander graphs, we also
prove that some of these metric quantities, or combinations thereof, do not to
deliver any spectral bounds with the correct scaling.
