I wrote some software in Matlab for this task for my own purposes, and it is available
for download from my website: Graph Manager. It was designed to work with Matlab 6.5 (the only version I own which was released over 10 years ago) and you should unzip it and run it with graphm from inside the created directory. It should look like this:

You can use the mouse to add/delete vertices and edges and do other things. Since it was for my use there are probably a few things that are unclear, feel free to ask here if there is a problem.
Since it was mainly used to look at graphs on the torus you will need to switch that off if you see the green box; you do that with "Save/Disp" and the "shape" submenu.
To get the adjacency matrix you can just go to the main Matlab window and it will be in the global variable A.