Whether you're considering a multigraph (which may have multiple edges and/or loops) or a  simple graph, both are CW complexes.  For any finite CW complex $G$, the *Euler characteristic* $\chi(G)$ is defined as the alternating sum (#0-cells)-(#1-cells)+(#2-cells)-...  (see [Wikipedia][1]). Thus for a finite graph, the Euler characteristic is $|V|-|E|$.  It's a homotopy invariant, and the operation of collapsing one edge and its vertices to a single vertex is a homotopy equivalence, so any function of $|V|-|E|$ is invariant under this operation.

When the graph is connected, the quantity $|E|-|V|+1$ ($=1-\chi(G)$) is the smallest number of edges that must be removed to yield a graph with no cycles, called the *cyclomatic number* or the *circuit rank* (see [Mathworld][2]).  But if the graph is not connected, then "$+1$" must be replaced by "$+k$," where $k$ is the number of components.


  [1]: http://en.wikipedia.org/wiki/Euler_characteristic
  [2]: http://mathworld.wolfram.com/CircuitRank.html