In computer science, one way of providing a semantics for programs is by using infinite trees to model do-loops in finite flow-graphs. Dana Scott and Marshall Hall are the earliest I recall. Arbib and Manes later on.
A lot of knot theory can be tackled by a similar tack.
For an introduction to asymptotic enumeration and random graphs (mentioned several times below), see:
- Edgar M. Palmer (1985), Graphical Evolution : An Introduction to the Theory of Random Graphs, John Wiley and Sons, New York, NY. http://portal.acm.org/citation.cfm?id=4050
For one of the inaugural applications of graph theory to social networks, see:
- Frank Harary, Robert Z. Norman, and Dorwin Cartwright (1965), Structural Models : An Introduction to the Theory of Directed Graphs, Wiley, New York, NY. http://www.ams.org/featurecolumn/archive/networks7.html
For applications to geography, there's this eBook:
- Arlinghaus, Sandra L.; Arlinghaus, William C.; Harary, Frank (2002), Graph Theory and Geography : An Interactive View, Wiley, New York, NY. http://deepblue.lib.umich.edu/handle/2027.42/58623
For recursive and self-similar graphs in knot theory, an ever-good springboard is:
- Louis H. Kauffman's home page : http://www.math.uic.edu/~kauffman/