2 Fixed typo: second instance of (#0-cells) ---> (#2-cells)

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)+(#0-cells)-... #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.

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)+(#0-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.