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.