I would like to know a good program for drawing graphs and analyzing them (finding Eulerian circuits, Hamilton cycles, etc.). I would also like to export the drawing to Word.
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Why don't you try Sage -- it's free and should do as good a job anything else. You just type in
for M an adjacency matrix to create your graph, then you can do
to find these things, and
draws your graph as a .png file which shouldn't be too hard to open in Word.