# Lawler-Kozdron-Richards-Stroock’s combined Proof for the Matrix-Tree theorem and Wilson’s Theorem

David Wilson and a cover of Shlomo’s recent book “Curvature in mathematics and physics”

A few weeks ago, in David Kazhdan’s basic notion seminar, Shlomo Sternberg gave a lovely presentation Kirchho ff and Wilson via Kozdron and Stroock. The lecture is based on the work presented in the very recent paper by Michael J. Kozdron,  Larissa M. Richards, and Daniel W. Stroock: Determinants, their applications to Markov processes, and a random walk proof of Kirchhoff’s matrix tree theorem. Preprint, 2013. Available online at arXiv:1306.2059.

Here is the abstract:

Kirchhoff’s formula for the number of spanning trees in a connected graph  is over 150 years old. For example, it says that if $c_2, \dots, c_n$ are the nonzero  eigenvalues of the Laplacian, then the number k of spanning trees is $k= (1/n)c_2\cdots c_n.$ There are many proofs.  An algorithm due to Wilson via loop erased random walks produces such a tree, and Wilson’s theorem is that all spanning trees are produced by his algorithm with equal probability. Hence,  after the fact, we know that Wilson’s algorithm produces any given tree with probability 1/k.  A proof due to Lawler, using the Green’s function, shows directly that Wilson’s algorithm has the probability 1/k  of producing any given spanning tree, thus simultaneously proving Wilson’s theorem and Kirchhoff’s formula. Lawler’s proof has been considerably simplified by Kozdron and Stroock. I plan to explain their proof. The lecture will be completely self-contained, using only Cramer’s rule from linear algebra.

(Here are also lecture notes of the lecture by Ron Rosenthal.)

Here is some background.

## The matrix-tree theorem

The matrix tree theorem asserts that the number of rooted spanning trees of a connected graph G  is the product of the non-zero eigenvalues of L(G), the Laplacian of G.

Suppose that G has n vertices. The Laplacian of G is the matrix whose (i,i)-entry is the degree of the ith vertex, and its (i,j) entry for $i \ne j$ is 0 if the ith vertex is not adjacent to the jth vertex, and -1 if they are adjacent. So  L(G)=D-A(G) where A(G) is the adjacency matrix of G, and D is a diagonal matrix whose entries are the degrees of the vertices.  An equivalent formulation of the matrix-tree theorem is that the number of spanning trees is the determinant of a matrix obtained from the Laplacian by deleting the j th row and j th column.

We considered a high dimensional generalization of the matrix tree theorem in these posts (I, II, III, IV).

## How to generate a random spanning tree for a graph G?

### Using the matrix-tree theorem

Method A: Start with an edge $e \in G$, use the matrix-tree theorem to compute the probability $p_e$ that e belongs to a random spanning tree of G, take e with probability $p_e$. If e is taken consider the contraction $G/e$ and if G is not taken consider the deletion $G \backslash e$ and continue.

This is an efficient method to generate a random spanning tree according to the uniform probability distribution. You can extend it by assigning each edge a weight and chosing a tree with probability proportional to the product of its weights.

### Random weights and greedy

Method B: Assign each edge a random real number between 0 and 1 and chose the spanning tree which minimizes the sum of weights via the greedy algorithm.

This is a wonderful method but it leads to a different probability distribution on random spanning trees which is very interesting!

### The Aldous-Broder random walk method

Method C: The Aldous-Broder theorem. Start a simple random walk from a vertex of the graph until reaching all vertices, and take each edge that did not form a cycle with earlier edges. (Or, in other words, take every edge that reduced the number of connected components of the graph on the whole vertex set and visited edges.)

Amazingly, this leads to a random uniform spanning tree. The next method is also very amazing and important for many applications.

### David Wilson’s algorithm

Method D: Wilson’s algorithm. Fix a vertex as a root. (Later the root will be a whole set of vertices, and a tree on them.) Start from an arbitrary vertex u not in the root and take a simple random walk until you reach the root. Next, erase all edges in cycles of the path created by the random walk so you will left with a simple path from  u to the root. Add this path to the root and continue!

Here is a link to Wilson’s paper! Here is a nice presentation by Chatterji  and Gulwani.

# A Proof by Induction with a Difficulty

The time has come to prove that the number of edges in every finite tree is one less than the number of vertices (a tree is a connected graph with no cycle). The proof is by induction, but first you need a lemma.

Lemma: Every tree with at least two vertices has a leaf.

A leaf is a vertex with exactly one neighbor.

Well, you start from a vertex and move to a neighbor, and unless the neighbor is a leaf you can move from there to a different neighbor and go on. Since there are no cycles and the tree is finite you must reach a leaf. Of course you describe the argument in greater detail and it seems that everybody is happy.

Good.

Theorem: A tree $T$  with $n$ vertices has $n-1$ edges.

After checking the basis for the induction we argue as follows: By the lemma $T$ has a leaf $v$ and once we delete $v$  and the edge containing it we are left with a graph $T'$ on $n-1$ vertices. Now, $T'$ has no cycles since $T$ did not have any. It takes some effort to show that $T'$ is connected. You describe it very carefully. Once this is done you know that $T'$ is a tree as well and you can apply the induction hypothesis.

Student: Now what about the case where $v$ is not a leaf? Continue reading

# A Beautiful Garden of Hypertrees

We had a series of posts (1,2,3,4) “from Helly to Cayley” on weighted enumeration of Q-acyclic simplicial complexes. The simplest case beyond  Cayley’s theorem were Q-acyclic complexes  with $n$ vertices, ${n \choose 2}$ edges, and ${{n-1} \choose {2}}$ triangles. One example is the six-vertex triangulation of the real projective plane. But here, as in many other places, we are short of examples.

Nati Linial,  Roy Meshulam and Mishael Rosenthal wrote a paper with very surprising examples of Q-acyclic simplicial complexes called “sum complexes”. The basic idea is very simple: The vertices are $\{1,2,\dots , n\}$. Next you pick three numbers $a,b$ and $c$ and consider all the triples $i,j,k$ so that $i+j+k$ is either $a$ or $b$ or $c$. And let’s assume that $n$ is a prime.

So how many triangles do we have? A fraction of $3/n$ of all possible triangles which is what we want (${{n-1} \choose {2}}$).

If the three numbers form an arithmetic progression then the resulting simplicial complex is collapsible. In all other cases it is not collapsible. The proof that it is Q-acyclic uses a result of Chebotarëv on Fourier analysis. (So what does Fourier analysis have to do with computing homology? You will have to read the paper!) The paper considers the situation in all dimensions.

What about such combinatorial constructions for Q-homology spheres?

# Helly’s Theorem, “Hypertrees”, and Strange Enumeration I

### 1. Helly’s theorem and Cayley’s formula

Helly’s theorem asserts: For a family of n convex sets in $R^d$, n > d, if every d+1 sets in the family have a point in common then all members in the family have a point in common.

Cayley’s formula asserts: The number of  trees on n labelled vertices is $n ^{n-2}$.

In this post (in two parts) we will see how an extension of Helly’s theorem has led to high dimensional analogs of Cayley’s theorem.

left: Helly’s theorem demonstrated in the Stanford Encyclopedia of Philosophy (!), right: a tree

### 2. Background

This post is based on my lecture at Marburg. The conference there was a celebration of new doctoral theses Continue reading