  We analyze the nonlinear optics of quasi one-dimensional quantum graphs and
manipulate their topology and geometry to generate for the first time
nonlinearities in a simple system approaching the fundamental limits of the
first and second hyperpolarizabilities. Changes in geometry result in smooth
variations of the nonlinearities. Topological changes between
geometrically-similar systems cause profound changes in the nonlinear
susceptibilities that include a discontinuity due to abrupt changes in the
boundary conditions. This work may inform the design of new molecules or nano-
scale structures for nonlinear optics and hints at the same universal behavior
for quantum graph models in nonlinear optics that is observed in other systems.
