What do graphene "armchair(5,5)" and "zigzag(5,0)" refer to?
The third picture, subpic (d) here shows you the geometric armchair(red) and zigzag(blue) structure:



It's derived from  structure description of Carbon nanotubes similar to surface reconstruction/overlayer structures:


  The integers n and m denote the number of unit vectors along two directions in the honeycomb crystal lattice of graphene. If m = 0, the nanotubes are called zigzag nanotubes, and if n = m, the nanotubes are called armchair nanotubes. Otherwise, they are called chiral. The diameter of an ideal nanotube can be calculated from its (n,m) indices


I hope you have some background, otherwise here is a nice visualisation, or look the lecture linked above. It's important to understand what the unit cell of a 3D crystal or 2D surface is.