edges minus vertices - MathOverflow

edges minus vertices

2010-09-28T17:38:12Z

Is there a more interesting name for this graph invariant? It seems to have been called 'complexity' here http://arxiv.org/abs/math/0502579 and here http://www.mathunion.org/ICM/ICM1994.2/Main/icm1994.2.1375.1383.ocr.pdf .

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.

Answer by Tony Huynh for edges minus vertices

2010-09-28T18:00:30Z

As Jack Lee mentions, the related quantity $|V|-|E|$ is often called the Euler characteristic of the graph. I have also heard $|E|-|V|+1$ called the cyclomatic number.

Answer by Louigi Addario-Berry for edges minus vertices

2010-09-28T18:03:54Z

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

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

Answer by Philippe Nadeau for edges minus vertices

2010-09-28T18:59:11Z

Cyclomatic number or excess are indeed common names for the quantity $|E|-|V|+1$ (as other answers mention). Let me add that the correct quantity to consider when you have $k$ components in your graph $G$ is $c(G):=|E|-|V|+k$. Then $c(G)=0$ means exactly that $G$ is a forest.
Also, the number $c(G)$ occurs in algebraic graph theory as the dimension of the cycle space of the graph $G$.

Answer by Jack Lee for edges minus vertices

2010-09-28T19:42:00Z

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.