  We dress bare quantum graphs with finite delta function potentials and
calculate optical nonlinearities that are found to match the fundamental limits
set by potential optimization. We show that structures whose first
hyperpolarizability is near the maximum are well described by only three
states, the so-called three-level Ansatz, while structures with the largest
second hyperpolarizability require four states. We analyze a very large set of
configurations for graphs with quasi-quadratic energy spectra and show how they
exhibit better response than bare graphs through exquisite optimization of the
shape of the eigenfunctions enabled by the existence of the finite potentials.
We also discover an exception to the universal scaling properties of the
three-level model parameters and trace it to the observation that a greater
number of levels are required to satisfy the sum rules even when the
three-level Ansatz is satisfied and the first hyperpolarizability is at its
maximum value, as specified by potential optimization. This exception in the
universal scaling properties of nonlinear optical structures at the limit is
traced to the discontinuity in the gradient of the eigenfunctions at the
location of the delta potential. This is the first time that dressed quantum
graphs have been devised and solved for their nonlinear response, and it is the
first analytical model of a confined dynamic system with a simple potential
energy that achieves the fundamental limits.
