Is there a more interesting name for this graph invariant: edges minus vertices? It seems to have been called 'complexity' in
- Remco van der Hofstad, Joel Spencer, Counting Connected Graphs Asymptotically, European Journal of Combinatorics 27 Issue 8 (2006) 1294–1320, doi:10.1016/j.ejc.2006.05.006, arXiv:math/0502579
and in
- Joel Spencer, Probabilistic Methods in Combinatorics In: Chatterji S.D. (eds) Proceedings of the International Congress of Mathematicians, Birkhäuser, Basel (1995), doi:10.1007/978-3-0348-9078-6_132, Wayback Machine pdf of article, big IMU pdf of whole volume
The motivation is that we want to talk about a quantity that is preserved under the graph transformation of collapsing two distinct vertices connected by an edge to a single vertex (thereby removing one edge and one vertex, preserving 'edges minus vertices'). So for example if the quantity 'edges minus vertices plus one' is more natural for some reason and has a name, then this would also be helpful. The concept should not be restricted to e.g. planar graphs.
Best Answer
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). 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). But if the graph is not connected, then "$+1$" must be replaced by "$+k$," where $k$ is the number of components.