# Total number of Spanning Trees in a Graph

If a graph is a complete graph with n vertices, then total number of spanning trees is n^{(n-2)} where n is the number of nodes in the graph. In complete graph, the task is equal to counting different labeled trees with n nodes for which have Cayley’s formula.

**What if graph is not complete?**

Follow the given procedure :-

STEP 1: Create Adjacency Matrix for the given graph.

STEP 2: Replace all the diagonal elements with the degree of nodes. For eg. element at (1,1) position of adjacency matrix will be replaced by the degree of node 1, element at (2,2) position of adjacency matrix will be replaced by the degree of node 2, and so on.

STEP 3: Replace all non-diagonal 1’s with -1.

STEP 4: Calculate co-factor for any element.

STEP 5: The cofactor that you get is the total number of spanning tree for that graph.

Consider the following graph:

Adjacency Matrix for the above graph will be as follows:

After applying STEP 2 and STEP 3, adjacency matrix will look like

The co-factor for (1, 1) is 8. Hence total no. of spanning tree that can be formed is 8.

NOTE- Co-factor for all the elements will be same. Hence we can compute co-factor for any element of the matrix.

This method is also known as Kirchhoff’s Theorem. It can be applied to complete graphs also.

Please refer below link for proof of above procedure.

https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem#Proof_outline

This article is contributed by **Kapil Khandelwal**. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

## Recommended Posts:

- Total number of Spanning trees in a Cycle Graph
- Program to find total number of edges in a Complete Graph
- Maximum Possible Edge Disjoint Spanning Tree From a Complete Graph
- Problem Solving for Minimum Spanning Trees (Kruskal’s and Prim’s)
- Maximum number of edges that N-vertex graph can have such that graph is Triangle free | Mantel's Theorem
- Number of Triangles in an Undirected Graph
- Number of sink nodes in a graph
- Count number of trees in a forest
- Number of trees whose sum of degrees of all the vertices is L
- Number of groups formed in a graph of friends
- Maximum number of edges in Bipartite graph
- Number of Simple Graph with N Vertices and M Edges
- Minimum number of edges between two vertices of a Graph
- Count number of edges in an undirected graph
- Minimum number of edges between two vertices of a graph using DFS