My answer comes from the random graphs community. In the book [Random Graphs][4], the quantity "edges minus vertices" is called the *excess*, which is quite standard terminology at least in random graphs.

In [these][1] [papers][2] we call the quantity "edges minus vertices plus one" the *surplus*. In an important paper in the area, [Aldous][3] calls edges beyond those in a spanning tree both surplus edges and excess edges. 

[1]: http://arxiv.org/abs/0903.4730
[2]: http://arxiv.org/abs/0908.3629
[3]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aop/1024404421
[4]: http://www.amazon.ca/Random-Graphs-Svante-Janson/dp/0471175412