How should I reinforce the center of my desk?
This sounds like a quick hack kind of desk -- no shame in that, but as you are finding out it has compromises.  Melamine is just the stiff plastic glued to the woody stuff underneath and provides no structural rigidity.  And the kinds of things melamine is glued to, particleboard and fiberboard, have very weak resistance to breakage.  I would not put a CRT monitor on a particleboard desk surface without reinforcement for example.

Strips of mild steel or hardwood screwed to the underside of the desk and running the entire long axis might help and be unobtrusive.  As an extra step, and one that also solves your edge chips, aluminum angle trim can be glued and screwed to the edges of the desk to reduce flex.  It won't be enough to prevent flex all by itself, but it will help.

If you need to build another desk like this, I would recommend 5/8" plywood as a superior alternative to particleboard, and real wood is best of all, as your finances and skill permits.