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

enter image description here

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?

enter image description here

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":

enter image description here

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

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    $\begingroup$ Why on earth do you think there is a relation between this graph and the torus? $\endgroup$ – Adam Smith Sep 20 '11 at 2:30
  • $\begingroup$ The graph consists of three triangles (top, bottom-left, bottom-right) arranged corner-wise in a circle. $\endgroup$ – Hans-Peter Stricker Sep 20 '11 at 9:21
  • $\begingroup$ It's the graph with vertices the 9 positions in a 3 by 3 array and adjacency defined by two positions being in the same row or column. $\endgroup$ – Hans-Peter Stricker Sep 20 '11 at 13:53
  • $\begingroup$ Is this graph even toroidal (embeddable on the torus)? It doesn't appear to be so. It's quite pretty, though. $\endgroup$ – Harry Stern Sep 20 '11 at 14:28
  • $\begingroup$ @Hans : Yes, I can see that. But what does that have to do with the torus? $\endgroup$ – Adam Smith Sep 20 '11 at 14:51
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The graph is indeed toroidal: toroidal embedding of given graph

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

three by three grid

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:

K7 on the torus

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  • $\begingroup$ What did you use to draw these pictures? $\endgroup$ – Harry Stern Sep 20 '11 at 15:49
  • $\begingroup$ I don't see how the second graph it's an embedding of the first, where are the triangles? $\endgroup$ – Giacomo d'Antonio Sep 20 '11 at 15:52
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    $\begingroup$ @Giacomo: The three vertices in each row or column form a triangle. $\endgroup$ – Rahul Sep 20 '11 at 16:00
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    $\begingroup$ @Harry, I draw my graphs on the torus using self-written Matlab code, available for download from my website at faculty.capebretonu.ca/jpreen/graphm.zip (You would need Matlab R13 or above to run the code, though) $\endgroup$ – jp26 Sep 20 '11 at 17:48
  • $\begingroup$ @James: Is "standard embedding" a defined term? Is there a standard embedding of $K_7$ on the torus? $\endgroup$ – Hans-Peter Stricker Sep 21 '11 at 7:39
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I found two interesting references for the "discrete torus":

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

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