what is the spanning tree ?

The spanning tree of a graph (G) is a subset of G that covers all of its vertices using the minimum number of edges.

Some properties of a spanning tree can be deduced from this definition:

1. Since “a spanning tree covers all of the vertices”, it cannot be disconnected.

2. A spanning tree cannot have any cycles and consist of $(n-1)$ edges (where $n$ is the number of vertices of the graph) because “it uses the minimum number of edges”.

A graph can have more than one spanning tree. In fact, a complete graph can have a maximum of n^{n-2}nn−2 spanning trees.