Closed curves on the discrete torus

Question

I came about the following graph which seems to me the smallest discrete version of the torus:

Is this graph treated under a special name? What can be said about its cycles? Can its cycles be grouped in some equivalence classes which can be related to homotopy classes of closed curves on the continuous torus?

Or are all cycles essentially the same on the discrete torus?

[Added] I came up with an even more intriguing - since more symmetric - picture of the "torus graph":

Maximal symmetry would be achieved only when the three vertices in the middle would coincide.

Answer 1

The graph is indeed toroidal:

Of course, Hans' graph also has a standard embedding too:

I would say that the graph which is the discrete version of the torus would be $K_7$, since it is a triangulation of the torus and also a vertex and edge transitive graph.

This is $K_7$ on the torus:

Answer 2

I found two interesting references for the "discrete torus":

• Bela Bollobas et al., Eliminating cycles in the discrete torus
• David Galvin, Sampling independent sets in the discrete torus

At least there is a thorough definition of the discrete torus.