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 caption reads:


  FIG. 1. (Color online) Geometry of the supercell used for our vdW DFT calculations. Upper part: 2 × 2 graphene lattice matched to √3 × √3 Ag(111) lattice. Lower part: Geometry of the supercell in perpendicular direction.


What exactly is a "√3 × √3 Ag(111) lattice" and how exactly does the square root of three come in to this?
"Root 3" or "$\sqrt{3}$" or "$\sqrt{3} \times \sqrt{3}$" are slang for the Wood's notation (1, 2) $ \left( \sqrt{3} \times \sqrt{3}\right)R30$, a commonly occurring configuration of one hexagonal 2D lattice with respect another.

In this example the substrate is the (111) face of silver. Since it is face-centered cubic with lattice constant of about 4.08 Angstroms the (111) surface is hexagonal with a lattice constant 4.08 / $\sqrt{2}$ = 2.88 Angstroms. 

The figure caption says "2 × 2 graphene lattice matched to √3 × √3 Ag(111) lattice."

The annotation in the OP's drawing copied below shows solid arrows for the Ag(111) lattice vectors the dashed arrows are rotated by 30 degrees and $\sqrt{3}$ longer, and they match the periodicity of a $2 \times 2$ graphene cell.

The lattice constant of free graphene is about 2.46 Angstroms and double that is 4.92. Compared to $\sqrt{3}$ times 2.88 = 4.99 Angstroms it's a fairly good lattice match.