Understanding Kirchhoff's theorem

Question

I am trying to the understand the Kirchhoff's theorem and it seems to be giving a hard time. Basically how do you explain why you have $n^{n-2}$ in finding the number of cycles in a given graph. A very basic explanation would be appreciated.

Answer 1

Cayley's formula counts the number of distinct labeled trees of a complete graph $K_n$. the number of spanning trees is equal to any cofactor of the Laplacian matrix of $K_n$. The Laplacian matrix is $$ \begin{bmatrix} n-1 & -1 & \cdots & -1 \\ -1 & n-1 & \cdots & -1 \\ \vdots & \vdots& \ddots & \vdots \\ -1 & -1 & \cdots & n-1 \\ \end{bmatrix}. $$ Any cofactor of the above matrix is $n^{n−2}$.