# Number of spanning trees of a weighted complete Graph

**Prerequisites:** Graph Theory Basics, Spanning tree.

**Complete Weighted Graph:** A graph in which an edge connects each pair of graph vertices and each edge has a weight associated with it is known as a complete weighted graph.

The number of spanning trees for a complete weighted graph with n vertices is **n ^{(n-2)}**.

**Proof: **Spanning tree is the subgraph of graph G that contains all the vertices of the graph. Therefore, the number of spanning trees of a complete weighted graph is the same as the number of labeled trees (need not be binary) with n vertices.

The Prüfer sequence of a labeled tree of n vertices is a unique sequence of length (n-2) associated with the tree. Also, for a given Prüfer sequence of length (n-2) on the labels 1 to n, there is a unique labeled tree with the given Prüfer sequence. Therefore, we have a **bijection** between the set A of **labeled trees with n vertices** and the set B of **Prüfer sequences of size n-2 on the labels 1 to n**. This can be proved as follows –

Let T be a labeled tree with vertices 1,2,…,n, and S as a Prüfer sequence of size (n-2). Thus, T and S are the elements of sets A and B, respectively.

**(i) Labeled tree (T) –> Prufer sequence (S)**

__Constructing the Prüfer sequence of a labeled tree __–

Initially, let S = NULL.

**Procedure –**

- Find the leaf node(L) of T with the smallest label.
- Add the neighbour of L to S.
- Delete the leaf node, L.
- Repeat the above steps until there are only two nodes left in the tree (Only one spanning tree is possible).
- We constructed the Prüfer sequence S associated with the labeled tree T.

**Observations –**

- No leaf node is appended to S.
- Every vertex V of tree T is added to S, a total of degree(V)-1 times.
- The tree T has n vertices and hence (n-1) edges.
- Number of terms in S = Sum of (degree(V) – 1) for all vertices V belonging to the tree T = Sum of degree of all vertices of tree T – (1+1+…+1..n times) = 2(no. of edges) – n = 2*(n-1) – n = n-2. (Since the sum of the degree of all vertices of a tree = 2*no. of edges of the tree).
- Therefore, T is analogous to the Prüfer’s sequence S of length (n-2).

**(ii) Prufer Sequence (S) –> Labeled Tree (T)**

__Constructing the labeled tree from its Pr__ü__fer sequence__–

**Procedure-**

- Let L = {1, 2, …, n} be the set of labels (vertices of T).
- Let S = {a
_{1},a_{2},…,a_{(n-2)}} be the Prüfer sequence of size (n-2) where each a_{i}belongs to L. - Find the smallest element x that belongs to L but is not in S.
- Connect x and the first element of S (a
_{1}) through an edge. - Delete a
_{1}from S, x from L (Thus, S:=S-{a_{1}} and, L:=L-{x}). - Similarly, find y, the smallest element belonging to L and not in S.
- Connect y and first element of S(a
_{2}). - Remove y from L and a
_{2}from S (Thus, S:=S-{a_{2}} and, L:=L-{y}). - Continue the above process till two items are left in L.
- Connect these two items in the tree formed so far.

The tree obtained from S is the same as T. Therefore, Prüfer sequence S of size (n-2) is analogous to T ( S <–> T ). Hence, there is a **bijection** between the set of labeled trees with n vertices and the set of Prüfer sequences of size (n-2) on the labels 1 to n.

Thus, the **number of spanning trees of a complete weighted graph of n vertices** = number of labeled trees with n vertices = number of Prüfer sequences of size (n-2) =** n ^{(n-2)}**.