Yes, certain restrictions are well known.  One reference is Kuperberg & Schramm <a href="http://arxiv.org/abs/math/9405218">here</a> ("Average kissing numbers for non-congruent sphere packings", 1994) which says that such graphs would have to have average degree <15.  A more recent reference is Benjamini & Schramm <a href="http://arxiv.org/abs/0910.3071">here</a> ("Lack of Sphere Packing of Graphs via Non-Linear Potential Theory", 2009) which shows that certain low degree infinite graphs are not realizable this way.