Edge Bounds and Degeneracy of Triangle-Free Penny Graphs and Squaregraphs - Eppstein, David (Journal of Graph Algorithms and Applications, 2018)
We show that triangle-free penny graphs have degeneracy at most two, and that both triangle-free penny graphs and squaregraphs have at most (2n-Ω(n),2n-D-2) edges, where n is the number of vertices and D is the diameter of the graph.