Recently I have studied Kirchhoff's spanning tree algorithm to count the number of spanning trees of a graph, which has the following steps:

  1. Build an adjacency matrix

  2. Replace the diagonal entries with the degrees of the corresponding nodes

  3. Replace all the other ones excluding the one's included in the
    diagonal by $-1$

  4. Then find the minor of any one of the elements to get your answer.

These all steps looks like magic to me. Can anyone explain how Kirchhoff derived this to obtain the number of spanning trees, I want to know the logic behind how this algorithm works, but all I can find in YouTube is the above steps only.


It looks like magic because the proof of Kirchhoff's Matrix Tree Theorem is nontrivial. It relies on several algebraic properties of the matrix constructed in steps 1-3, which is called the Laplacian matrix of the graph. Let $A$ be the adjacency matrix, and let $D$ be the diagonal matrix with the degrees of the nodes on the diagonal. Steps 1-3 build the matrix $L = D-A$: this is the Laplacian matrix of the graph.

So the question becomes, why should a minor of the Laplacian matrix be equal to the number of spanning trees of the graph?

To answer this question, we need to use other properties of $L$. One property is that we can also write $L$ as $B \cdot B^\top$ where $B$ is the oriented incidence matrix of the graph, that is, a matrix with one row per node $u$ and one column per edge $e$, where entry $B_{ue}=1$ if $e=\{u,v\}$ with $u<v$, $B_{ue}=-1$ if $e=\{u,v\}$ with $v < u$, and $B_{ue}=0$ if $e$ does not contain $u$. For example, for a triangle graph, the Laplacian matrix is $$ L=\left[\begin{array}{ccc} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{array}\right] $$ and the incidence matrix is $$ B=\left[\begin{array}{ccc} 1 & 1 & 0 \\ -1 & 0 & 1 \\ 0 & -1 & -1 \end{array}\right]. $$ Note that the inner product of two distinct rows of $B$ is $-1$ if the corresponding nodes are adjacent, and 0 if they are not. The inner product of a row of $B$ with itself equals the degree of the corresponding node. (Just use the definition of $B$.) This is why $L=B\cdot B^\top$.

Assuming the graph is connected, the rank of $B$ is exactly $n-1$ where $n$ is the number of nodes of the graph. It is not $n$ because if you sum all rows, you get a zero vector. It is $n-1$ because the number of independent columns of $B$ is $n-1$: if you take any subset of columns corresponding to the set of edges of any spanning tree of the graph, these columns will be independent; the only way to have a dependent set of columns is when the edges selected contain a cycle.

If we delete a row and corresponding column of $L$, we obtain a matrix $M$. If we delete the corresponding row of $B$, we obtain a matrix $C$. Now $M=C \cdot C^\top$ for the same reasons that $L=B \cdot B^\top$. We are interested in $\det(M)=\det(CC^\top)$. In the example of the triangle graph above, $$ M=\left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right] $$ and $$ C=\left[\begin{array}{ccc} 1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right]. $$

The last step is to invoke another algebraic fact, the Cauchy-Binet formula. This says that for any matrix $C$,
$$ \det(C C^\top) = \det (C_{K_1}) \det(C_{K_1}^\top) + \ldots + \det (C_{K_q}) \det(C_{K_q}^\top) $$ where each $C_{K_i}$ denotes an $(n-1) \times (n-1)$ submatrix of $C$ and the sum runs over all such submatrices. Thus, $$ \det(C C^\top) = \det (C_{K_1})^2 + \ldots + \det (C_{K_q})^2. $$

However, $\det(C_{K_i})=\pm 1$ if the edges corresponding to $C_{K_i}$ comprise a spanning tree, while $\det(C_{K_i})=0$ otherwise (because then the set contains a cycle and the corresponding columns are not independent). Thus, the value of the sum is exactly the number of spanning trees.


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