Curious about Symmetric graphs of regular simplices Found this picture on wikipedia: http://en.wikipedia.org/wiki/Simplex#Symmetric_graphs_of_regular_simplices
Can anyone explain why some of these graphs contain their (geometric) center, and others don't? At first I thought it had to do with the simplex being prime dimension, but then I noticed that it is also true about 9. Does this correspond to some interesting number theoretic or group theoretic property? Or is this a superficial observation?
Thank you.

Actually it's rather simple, the symmetric graphs with odd vertex valence include their center precisely because there are an even number of vertices evenly distributed on a circle, and so each vertex has a corresponding vertex antipodal to it - it follows that these two points have an edge between them which goes through the geometric center.
Similarly, for even valence there are an odd number of vertices evenly distributed on the circle and so no antipodal pairs exist. Hence no edges form a diameter which would require end points to be an antipodal pair.
