If you are looking at general approaches to problems like these, you probably want to get some sort of computer package that does these kinds of computations. It would be similar to something like Maple, Mathematica, or MATLAB, but designed mainly for graphs and graph-theoretic problems.
If you are familiar with programming and software development, you will want to look at things like this:
List of C++ libraries for Graph Theory - Stack Overflow
If you want a nice package with editor, GUI, and good scripting language then something like this (which is a part of Mathematica).