Understanding lattice matching of graphene on silver (111) crystal surface and meaning of the √3 x √3 lattice?

Figure 1 in Ivor Lončarić and Vito Despoja, Phys. Rev. B 90, 075414 Benchmarking van der Waals functionals with noncontact RPA calculations on graphene-Ag(111) (available in Researchgate) shows a schematic of the graphene structure on what I believe is a silver (111) surface.

• The $\sqrt{3}$ is the distance shown as black lines between every other C atom (assuming bond length is one unit). Twice this is the equivalent length between the Ag atoms shown as silver/grey. So I'm assuming this is what they mean by lattice matching? – porphyrin Jun 10 at 10:23
• Start at the left bottom most blue atom (crossed by 2 black lines), go north by 1 atom and north east by 1 atom. The distance as the crow flies is $\sqrt{3}$ times bond length, i.e. length of black line connecting atoms. Now look at underlying Ag atoms; twice this length is equivalent length in this lattice so I guessed that this is connection between lattices and why one fits over the other. This could be rubbish, however, as I'm not an expert in surface science. – porphyrin Jun 12 at 11:12