There are some [url=http://www2.stetson.edu/~efriedma/papers/matchstick.ppt]slides by Erich Friedman[/url] on the topic. The slides also show the [url=http://mathworld.wolfram.com/HarborthGraph.html]Harborth graph[/url] depicted in your copy of Number Freak. Using the string shown at the bottom of page 22 of Friedman's slides, you could elegantly build a 4-regular matchstick graph with 3*21 = 63 vertices and 126 edges. (page 28 contains it as a subgraph.) Yet Harborth's solution is the smallest known 4-regular matchstick graph with 52 vertices and 104 edges.