X-Nico

Euclidean minimum spanning tree

The simplest algorithm to find an EMST in two dimensions, given n points, is to actually construct the complete graph on n vertices, which has n(n-1)/2 edges, compute each edge weight by finding the distance between each pair of points, and then run a standard minimum spanning tree algorithm (such as the version of Prim's algorithm or Kruskal's algorithm) on it.

Kirchhoff's theorem

It is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph.

Minimum spanning tree

Alan M. Frieze showed that given a complete graph on n vertices, with edge weights that are independent identically distributed random variables with distribution function $F$ satisfying $F\text{'}(0)\; >\; 0$, then as n approaches +∞ the expected weight of the MST approaches $\backslash zeta(3)/F\text{'}(0)$, where $\backslash zeta$ is the Riemann zeta function.

see also

Kirchhoff's theorem

Cayley's formula follows from Kirchhoff's theorem as a special case, since every vector with 1 in one place, −1 in another place, and 0 elsewhere is an eigenvector of the Laplacian matrix of the complete graph, with the corresponding eigenvalue being n.

Topological graph

The prototype of such results is Turán's theorem, where there is one forbidden subgraph: a complete graph with k vertices (k is fixed).