What is the number of spanning trees in an undirected simple graph?

My attempt:

Let $m$ be the number of edges in a simple graph, and let $n$ be the number of vertices.

Then number of spanning trees is $\binom{m}{n-1}$ minus the number of cycles of length $n-1$.

I read on Wikipedia that the number of spanning trees in the complete graph $K_n$ is $n^{n-2}$.

According to the formula I stated above, it should be $\binom{n(n-1)/2}{n-1} - \binom{n}{n-1} (n-1)!$.

How do I show that this is equal to $n^{n-2}$ for $K_n$?


You cannot show this since it isn't true. I encourage you to try out some actual numbers (e.g. $n=2$) and see for yourself that the numbers don't match. The problem is that what you should be subtracting is not the number of cycles of length $n-1$, but rather the number of collections of $n-1$ edges which contains at least one cycle. Such a collection need not be a cycle of length $n-1$.

In addition, the formula for the number of cycles of length $n-1$ is wrong. The number is $n(n-2)!$ rather than your $n(n-1)!$, and even this formula is only valid for $n \geq 4$ (for smaller $n$ there are no such cycles). However, even after replacing $n(n-1)!$ by $n(n-2)!$, the numbers don't match.

  • $\begingroup$ sir i got what you are saying. Can you please elaborate how you wrote formula for number of cycles of length n-1: What i did was --> choose n-1 vertices out of n vertices. Cycle is permutation of n-1 vertices. SO i thought it was nC(n-1)*(n-1)! . I have recently started studying graph theory. I was just thinkng about this problem. I saw some proofs based on kirchofs theorem etc which was way above my head. How the equivalence between that laplacian matrix and number of trees was established ? $\endgroup$ – Nascimento de Cos Nov 10 '19 at 15:42
  • $\begingroup$ I wrote wrongly. It should be number of collections of size n-1 with cycle in it. $\endgroup$ – Nascimento de Cos Nov 10 '19 at 15:45
  • $\begingroup$ The matrix tree theorem is a standard result, and there are many lecture notes and textbooks covering it. If it’s above your head then you’ll need to step up to it. $\endgroup$ – Yuval Filmus Nov 10 '19 at 17:24
  • $\begingroup$ Thank you sir, i will refer to some lecture notes. $\endgroup$ – Nascimento de Cos Nov 10 '19 at 19:08

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