I am trying to solve some basic graph theory problem and I want to know how many seven edges connected graphs are there. I think the exact number is 79 but I am not sure. For example for the case of six edges I have obtained 30 connected graphs. Thank you for your helps.
This seems correct. The numbers are 4, 19, 33, and 23 graphs on 5, 6, 7, and 8 vertices respectively.
Not too hard to verify using SageMath via the following code:
graph_list= # number of vertices is at least 5 and at most 8. for n in range(5,9): for g in graphs(n): if g.is_connected() and g.num_edges()==7: graph_list.append(g) print len(graph_list) # For some pictures! :-) graphs_list.show_graphs(graph_list)
This is a brute force approach and takes a bit more than a minute to run at the sagecell server.