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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

G = Graph (M)

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.

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You can also use sage's notebook interface to "draw" the graph. For this you can use the command graph_editor. – fidbc Apr 24 '13 at 21:29
The sage wiki also hosts this nice list of graph theory software. It might be a bit outdated but it is comprehensive. – fidbc Apr 24 '13 at 21:35

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