The one software I would not like to miss for drawing and editing graphs is yEd, a free software provided by yWorks. It is an intuitive tool to draw small graphs and networks where - in contrast to most normal vector graphics programs - moving around the nodes will automatically include the edges as well. Furthermore it is possible to mark a subgraph and to turn it or to mirror its position - without mirroring or turning the node labels as well.
I've created almost all drawings for my publications with it, especially those for my book and this blog. My normal design pipeline is to draw the graph in yEd and then either to save it directly as eps or pdf for inclusion in LaTeX documents or to first save it as an svg. Then I would give some finishing touches to the drawing in inkscape, from which it can then again be exported as eps and pdf.
Despite the fact that I've been using this tool now for more than 12 years, I just learned that it is also the best way to create an image for Power Point presentations. Instead of exporting it to bmp which scales badly, try exporting it to emf. Scaling this format leads to much better results.