Editing Kirchhoff's theorem (section)

=== Cayley's formula ===
{{main|Cayley's formula}}
[[Cayley's formula]] follows from Kirchhoff's theorem as a special case, since 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 (vector space)|dimension]] ''n''&nbsp;−&thinsp;1, so there are no other non-zero eigenvalues.

Alternatively, note that as Cayley's formula gives the number of distinct labeled trees of a complete graph ''K<sub>n</sub>'' we need to compute any cofactor of the Laplacian matrix of ''K<sub>n</sub>''. The Laplacian matrix in this case is
:<math>\begin{bmatrix}
  n-1 & -1      & \cdots & -1      \\
  -1  & n-1     & \cdots & -1      \\
  \vdots & \vdots& \ddots & \vdots \\
  -1 & -1      & \cdots & n-1      \\
\end{bmatrix}.</math>

Any cofactor of the above matrix is ''n<sup>n</sup>''<sup>−2</sup>, which is Cayley's formula.