Graham and Harary [3]studied how the diameter of hypercubes can be affected by increasing and decreasing edges. Since many networks are constructed by graph products, e.g., tori and meshes, in this paper we study how the diameter of graph products, particularly Cartesian product and Kronecker product, can be changed by adding and removal of edges. We study Cartesian products on paths, cycles, trees, and hypercubes. The diameter of the Kronecker product of two graphs is in general difficult to find. We in particular study the Kronecker product of two cycles.