Impediments to diffusion in quantum graphs: Geometry-based upper bounds on the spectral gap

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