Special issue on Selected papers from the Twenty-fifth International Symposium on Graph Drawing and Network Visualization, GD 2017 Edge Bounds and Degeneracy of Triangle-Free Penny Graphs and Squaregraphs Vol. 22, no. 3, pp. 483-499, 2018. Regular paper. Abstract We show that triangle-free penny graphs have degeneracy at most two, and that both triangle-free penny graphs and squaregraphs have at most $\min\bigl(2n-\Omega(\sqrt n),2n-D-2\bigr)$ edges, where $n$ is the number of vertices and $D$ is the diameter of the graph. Submitted: October 2017. Reviewed: January 2018. Revised: February 2018. Accepted: February 2018. Final: March 2018. Published: September 2018. Communicated by Fabrizio Frati and Kwan-Liu Ma article (PDF) BibTeX