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Hello everyone interested. Here's a problem I can't figure out. I wish to find the number of spanning trees in a simple, complete graph G with at least three vertices after I remove one of its edges.

I tried counting but even for small values of n this becomes hard. Is there a theorem or some other result I could employ?

Thanks in advance...

MelaniesWoes
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    Assuming you know how many spanning trees there are in the complete graph, you just need to subtract the ones that contain the missing edge. – Michael Biro May 09 '18 at 00:08

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Hint: The number of spanning trees without removing the edge is given by the Cayley formula. Given that a tree contains exactly $n-1$ edges, and given that the problem is symmetric (it doesn't matter which edge you remove), you can count in two ways the number of elements in the set $\{(e,T): e \text{ is an edge}, T \text{ is a tree containing }e\}$.

Arnaud Mortier
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  • How? Can you provide me with an example please? – MelaniesWoes May 09 '18 at 00:12
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    @Melanie sorry, my previous idea was a little complicated, I found something better. – Arnaud Mortier May 09 '18 at 00:22
  • Since the graph is complete, the edges are n choose 2, right? But how does that help...? Ps. I do note this is the uncomplicated version, haha! – MelaniesWoes May 09 '18 at 00:28
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    @Melanie Yes, $n$ choose $2$ will be helpful. So once you have chosen an edge (where you had $n\choose 2$ options), how many trees are there containing that edge? That's the very number you want to compute! So there you are, first counting, the cardinality of our set is ${n\choose 2}\times x$, where $x$ is the number you are looking for. – Arnaud Mortier May 09 '18 at 00:51
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    @MelaniesWoes to be more precise the number you are looking for is the total number of trees minus $x$. – Arnaud Mortier May 09 '18 at 00:56
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It is not hard to prove by induction on the number of components (using prufer codes and newtons formula) that if $G$ is a forest with components of size $c_1,c_2,\dots , c_k$ then the number of spanning trees containing $G$ is $n^{k-2}\prod\limits_{i=1}^n c_i$.

In your particular case the forest has $n-1$ components of sizes $2,1,1,1\dots$

And so the number of trees not containing the edge is $n^{n-2} -2n^{n-3}$

Asinomás
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