Bug in Graph.girth in 4.7.2 ?
sage 4.7.2 incorrectly gives girth of 4 on this graph:
sage: H=Graph([(0, 1), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 6), (2, 5), (3, 4), (5, 6)])
sage: H.girth()
4
There is a cycle (0,1,3) and sage 4.3 and magma correctly return 3.
I copied your code and Sage 4.6 returns 3.