number of vertices - number of edges + number of faces

This is called Euler's formula

Intro, relating it to the Gauss-Bonnet theorem

It is a Topological invariant.

For a CW complex, it is defined as the number of even-dimensional cells minus the number of odd-dimensional cells. the Euler characteristic of a cell complex depends only on its homotopy type, so the fact that the house with two rooms (from Hatcher) has the homotopy type of a point implies that its Euler characteristic must be 1, no matter how it is represented as a cell complex.

is since m S 1 has Euler characteristic 1−m . But it is a rather nontrivial theorem that the Euler characteristic of a space depends only on its Homotopy type

Euler's formula for triangle meshes

Each edge is adjacent to two faces. Each face has three edges. Think of half-edges, etc. For a closed surface, every face has 3 half-edges, so it has 3/2 edges, so $2E = 3F$. Often $E, F, V \gg \chi$, so that we can use Euler's formula to find that $F \approx 2V$, and $E \approx 3V$. Average valence of a vertex $\approx 3$

See here