How Many Spanning Trees Can Be Formed in an Extended n-Cycle Graph?

[squaremeplz]

Homework Statement

Take an n-cycle and connect two of its nodes at distance 2 by an edge. Find the number of spanning trees in this graph.

Homework Equations

n^(n-2)?

The Attempt at a Solution

I honestly have no idea on how to start this problem. I know the definition of an n-cycle. My approach towards problems like this would be to construct a graph. For example a cube is a 4-cycle with nodes 1,2,3,4. then connect nodes 1 and 3 which gives the distance of 2. This graph has 8 spanning trees. I counted them. But I have no idea what the general formula is for this. Any input would be much appreciated!

 

[mighty2000]

Hello, thank you for bringing up this interesting problem! Let's break it down step by step.

First, let's define an n-cycle as a graph with n vertices arranged in a circular pattern, where each vertex is connected to the two vertices adjacent to it. For example, a 4-cycle would have vertices 1, 2, 3, 4 and edges (1,2), (2,3), (3,4), (4,1).

Now, let's consider connecting two nodes at distance 2 by an edge. In the example of the 4-cycle, this would mean connecting nodes 1 and 3. This creates a new edge (1,3) and also creates a new path from node 1 to node 3 through the edge (2,4).

To find the number of spanning trees in this new graph, we can use the Matrix Tree Theorem. This theorem states that the number of spanning trees in a graph is equal to the determinant of any cofactor of the Laplacian matrix of the graph.

In this case, the Laplacian matrix for our new graph would be:

| 3 -1 -1 0 |
| -1 2 0 -1 |
| -1 0 2 -1 |
| 0 -1 -1 2 |

To find the determinant, we can use the cofactor expansion method along the first row. This gives us:

det(L) = 3 * det(2 -1 -1 | 0 2 -1 | -1 -1 2) - (-1) * det(-1 0 -1 | 0 2 -1 | -1 -1 2) = 3 * 3 - (-1) * (-3) = 12

Therefore, the number of spanning trees in this graph is 12.

In general, the formula for the number of spanning trees in a graph with n vertices and m edges is given by n^(n-2-m). In this case, n=4 and m=4, so the formula gives us 4^(4-2-4) = 4^(-2) = 1/16. However, as we can see from our example, this formula does not take into account the specific arrangement of edges in the graph. Therefore, the Matrix Tree Theorem is a more accurate method for finding the number of