Talk:Kirchhoff's theorem

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I copied the original material from spanning tree (mathematics) and added a different formulation of Kirchhoff's theorem using the cofactor of the admittance matrix. Then I wrote a new article admittance matrix and deleted most of the explaination on this page in terms of discrete Laplace operator (which was just a definition of the admittance matrix). Now the article is quite terse, perhaps someone else can add some material to illustrate the theorem. Although I do not think it is a good idea to duplicate the definition of the admittance matrix here. MathMartin 17:32, 30 Jan 2005 (UTC)

hmm?

Fair enough we an make a spanning tree, a loop free mechanism to determine the nodes, we can also determine the number of spanning trees in the graph (logically if there are k Vertices in the graph there is always a way to connect all k via some loop free path) however i doubt how it helps to say that path 102 is the same as path 201

Cayleys formula

Seeing that Cayley's formula follows from Kirchhoff's theorem as a special case is easy: every vector with 1 in one place, -1 in another place, and 0 elsewhere is an

eigenvector of the Laplacian matrix of the complete graph, with the corresponding eigenvalue being n. These vectors together span a space of dimension n-1, so there

are no other non-zero eigenvalues.

I don't think this is fully correct. First it should be mentioned that we're talking about $M_{11}$ , not $Q$ . $M_{11}$ is $n-1\times n-1$ and has $n-2$ eigenvectors of the form $(\dots ,0,1,0,\dots ,0,-1,0,\dots )$ to the eigenvalue $n$ and one eigenvector $(1,1,\dots ,1)$ to the eigenvalue 1. Multiplying those eigenvalues gives obviously $n^{(}n-2)$ . —Preceding unsigned comment added by 134.169.77.186 (talk) 13:33, 14 November 2008 (UTC)