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 n − 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_n we need to compute any cofactor of the Laplacian matrix of K_n. The Laplacian matrix in this case is

${\displaystyle {\begin{bmatrix}n-1&-1&\cdots &-1\\-1&n-1&\cdots &-1\\\vdots &\vdots &\ddots &\vdots \\-1&-1&\cdots &n-1\\\end{bmatrix}}.}$

Any cofactor of the above matrix is nⁿ⁻², which is Cayley's formula.

Formulation using eigenvalues of the Laplacian matrix

The theorem can be formulated in terms of the eigenvalues of the Laplacian. These eigenvalues are always non-negative and one of them is zero. For a given graph G with n vertices, let 0 = λ₀ ≤λ₁ ≤ λ₂ ≤...≤ λ_n−1 be the eigenvalues of its Laplacian matrix. Then the number t(G) of spanning trees of G is

${\displaystyle t(G)={\frac {1}{n}}\lambda _{1}\lambda _{2}\cdots \lambda _{n-1}\,.}$

Kirchhoff's theorem for multigraphs

Kirchhoff's theorem holds for multigraphs as well; the matrix L is modified as follows:

• The entry q_i,j equals −m, where m is the number of edges between i and j;
• when counting the degree of a vertex, all loops are excluded.

Cayley's formula for a complete multigraph is mⁿ⁻¹(nⁿ⁻¹−(n−1)nⁿ⁻²) by same methods produced above, since a simple graph is a multigraph with m = 1.

Explicit enumeration of spanning trees

Kirchhoff's theorem can be strengthened by altering the definition of the Laplacian matrix. Rather than merely counting edges emanating from each vertex or connecting a pair of vertices, label each edge with an indeterminate and let the (i, j)-th entry of the modified Laplacian matrix be the sum over the indeterminates corresponding to edges between the i-th and j-th vertices when i does not equal j, and the negative sum over all indeterminates corresponding to edges emanating from the i-th vertex when i equals j.

The determinant of the modified Laplacian matrix by deleting any row and column (similar to finding the number of spanning trees from the original Laplacian matrix), above is then a homogeneous polynomial (the Kirchhoff polynomial) in the indeterminates corresponding to the edges of the graph. After collecting terms and performing all possible cancellations, each monomial in the resulting expression represents a spanning tree consisting of the edges corresponding to the indeterminates appearing in that monomial. In this way, one can obtain explicit enumeration of all the spanning trees of the graph simply by computing the determinant.

For a proof of this version of the theorem see Bollobás (1998).^[3]

Matroids

The spanning trees of a graph form the bases of a graphic matroid, so Kirchhoff's theorem provides a formula for the number of bases in a graphic matroid. The same method may also be used to determine the number of bases in regular matroids, a generalization of the graphic matroids (Maurer 1976).

Kirchhoff's theorem for directed multigraphs

Kirchhoff's theorem can be modified to give the number of oriented spanning trees in directed multigraphs. The matrix Q is constructed as follows:

• The entry q_i,j for distinct i and j equals −m, where m is the number of edges from i to j;
• The entry q_i,i equals the indegree of i minus the number of loops at i.

The number of oriented spanning trees rooted at a vertex i is the determinant of the matrix gotten by removing the ith row and column of Q

Counting spanning k-component forests

Kirchhoff's theorem can be generalized to count k-component spanning forests in an unweighted graph.^[4] A k-component spanning forest is a subgraph with k connected components that contains all vertices and is cycle-free, i.e., there is at most one path between each pair of vertices. Given such a forest F with connected components ${\textstyle F_{1},\dots ,F_{k}}$, define its weight ${\textstyle w(F)=|V(F_{1})|\cdot \dots \cdot |V(F_{k})|}$ to be the product of the number of vertices in each component. Then

${\displaystyle \sum _{F}w(F)=q_{k},}$

where the sum is over all k-component spanning forests and ${\textstyle q_{k}}$ is the coefficient of ${\textstyle x^{k}}$ of the polynomial

${\displaystyle (x+\lambda _{1})\dots (x+\lambda _{n-1})x.}$

The last factor in the polynomial is due to the zero eigenvalue ${\textstyle \lambda _{n}=0}$. More explicitly, the number ${\textstyle q_{k}}$ can be computed as

${\displaystyle q_{k}=\sum _{\{i_{1},\dots ,i_{n-k}\}\subset \{1\dots n-1\}}\lambda _{i_{1}}\dots \lambda _{i_{n-k}}.}$

where the sum is over all (n − k)-element subsets of ${\textstyle \{1,\dots ,n\}}$. For example

${\displaystyle {\begin{aligned}q_{n-1}&=\lambda _{1}+\dots +\lambda _{n-1}=\operatorname {tr} Q=2|E|\\q_{n-2}&=\lambda _{1}\lambda _{2}+\lambda _{1}\lambda _{3}+\dots +\lambda _{n-2}\lambda _{n-1}\\q_{2}&=\lambda _{1}\dots \lambda _{n-2}+\lambda _{1}\dots \lambda _{n-3}\lambda _{n-1}+\dots +\lambda _{2}\dots \lambda _{n-1}\\q_{1}&=\lambda _{1}\dots \lambda _{n-1}\end{aligned}}}$

Since a spanning forest with n−1 components corresponds to a single edge, the k = n − 1 case states that the sum of the eigenvalues of Q is twice the number of edges. The k = 1 case corresponds to the original Kirchhoff theorem since the weight of every spanning tree is n.

The proof can be done analogously to the proof of Kirchhoff's theorem. An invertible ${\displaystyle (n-k)\times (n-k)}$ submatrix of the incidence matrix corresponds bijectively to a k-component spanning forest with a choice of vertex for each component.

The coefficients ${\textstyle q_{k}}$ are up to sign the coefficients of the characteristic polynomial of Q.