What software/site to use for drawing graphs (with vertices and edges, not of functions)?

I use tikz. It is a latex package.
I use tikz (latex) and OmniGraffle on Mac. The later tends to be faster if you need to through together a nice looking graph on a short fuse while the former gives you a lot of control.
I personally use Inkscape for most of my diagrams, and LibreCAD if I want to make something a bit more complicated (which isn't the use case here)

By using Inkscape and terminating a line in a circle you can represent a graph, you also have arrow terminations for directed graphs, and the circle tool works for nodes explicitly adjacent to themselves, diagrams look great and are saved in vector format
If you just search for a quick and easy way i recommend ipe. It lets you place your nodes and edges on a grid and you can export it as a vector image or pdf which you can import in your Latex file.
An easier and more visual but less cool approach - You can use a geometry tool like Geogebra, and drawing it as you would by hand on a paper
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In case you are not looking for a programmatic tool, but a WYSIWYG tool for drawing, Inkscape is nice for making SVG images, which can be exported to pdf and used in latex.

edit: it was already mentioned.. okey...
I use inkscape. At first I used tikz but I got frustrated, I think inkscape is more flexible (and learning to use it is more useful in general).
The network theory professors in my alma mater heavily encourage Gephi and NetworkX. If you just want to visualize some graphs, that's easy. Don't reinvent the wheel.

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