On 2D Inverse Problems/Special matrices

An important object containing information about a weighted graph G(V,E,w) is its Laplacian matrix. Given a numbering of the vertices of the graph, it's an n by n square matrix L_G, where n is the number of vertices in the graph, with the entries:

$l_{kl}:= \begin{cases} \sum_{v_k \rightarrow v_l} w_{kl} \mbox{ if}\ k = l, \\ -w_{kl} \mbox{ if}\ v_k \rightarrow v_l, \\ 0, \mbox{ otherwise,} \end{cases}$

where v_k → v_l means that there is a directed edge from vertex v_k to the vertex v_l, and where w is the weight function.

Exercise (*). Given a directed graph G without cycles, prove that one can number its vertices so that the corresponding Laplacian matrix L_G is triangular.

Given a weighted graph with boundary it is often convenient to number its boundary vertices first and to write its Laplacian matrix in the block form.

$L_G = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$

Exercise (*). Prove that a function/vector u is harmonic at the interior nodes of a graph G if

$u|_{int G} = -D^{-1}Cu|_{\partial G}.$

Let

$M=\begin{pmatrix} A & B \\ C & D \end{pmatrix}$

be a block matrix with an invertible square block D.

Then the Schur complement of the matrix M w/respect of the block D is the matrix

$M/D = A-BD^{-1}C.$

The Leibniz definition of determinant (a multilinear function in the rows and columns) of a square n by n matrix M is:

$\det M = \sum_{\sigma \in S_n} \sgn(\sigma) \prod_{k=1}^n M_{k,\sigma_k}.$

Exercise (*). Prove the following determinant identity for a square matrix M:

$\det M = \det(M/D) \det D.$

The following matrix W(G) consisting of random walk exiting probabilities (sums over weighted paths in a graph) plays an important role as boundary measurement for inverse problems. Suppose a weighted graph G has N boundary nodes, then the kl 'th entry of the N by N matrix equals to the probability that the first boundary vertex, that a particle starting its random walk at the boundary vertex v_k occupies, is the boundary vertex v_l. For a finite connected graph the columns of the matrix W(G) add up to 1.

Exercise (**). Derive an explicit formula for the matrix W(G) in terms of the blocks of Laplace matrix L(G) of the graph G.

$W_G = I - D_A^{-1}(A - BD^{-1}C).$

Exercise (***). Prove the following expansion formulas for entries and blocks of the matrix W(G),

• for two boundary vertices p_k and p_l of a graph G

$w_{kl} = \sum_{v_k\xrightarrow[]{path} v_l}\prod_{e=(p,q)\in path}w(e)/l_{pp},$

• for two distinct boundary vertices v_k and v_l of a graph G

$w_{kl}\det D = \frac{1}{l_{kk}}\sum_{v_k\xrightarrow{path}v_l}\prod_{e\in path}l(e)\det D(\tilde{path},\tilde{path}),$

where

$\tilde{path} = V-(\partial G\cup path).$

• for two disjoint subsets of boundary vertices P and Q of size n of a graph G, see [6],[7] and [14]

$\det W(P,Q) \det D = \pm\frac{\sum_{\sigma\in S_{n}}(-1)^{\sgn(\sigma)}\sum_{p_k\xrightarrow{paths}q_{\sigma_k}}\prod_{e\in paths}l(e)\det D(\tilde{paths},\tilde{paths})}{\prod_{p\in P}l_{pp}},$

where

$\tilde{paths} = V-(\partial G\cup paths).$

The exercises above provide a bridge b/w connectivity property of graph G and ranks of submatrices of its Laplacian matrix L(G) and the matrix of hitting probabilities W(G).

Exercise (*). Let G be a planar graph w/natural boundary, numbered circulary. Let P and Q be two non-interlacing subsets of boundary nodes of size n. Prove that

$(-1)^{\frac{n(n+1)}{2}}\det\Lambda_G(P,Q) \ge 0,$

w/the strict inequality iff there is a disjoint set of paths from P to Q.

Exercise (*). Show that the numbers of paths in the following graph are equal to the binomial coefficients.

Gluing the graphs w/out loops corresponds to multiplication of the weighted paths matrices.

Exercise (**). Use the result of the previous exercise to it to prove the following Pascal triangle identity, see[13],

$\begin{pmatrix} 1 & 1 & 1 & 1 & \ldots \\ 1 & 2 & 3 & 4 & \ldots \\ 1 & 3 & 6 & \ldots & \ddots \\ 1 & 4 & 10 & \ldots & \ddots \\ 1 & \ldots & \ldots & \ddots & \ddots \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 & \ldots \\ 1 & 1 & 0 & 0 & \ldots \\ 1 & 2 & 1 & 0 & \ddots \\ 1 & 3 & 3 & \ldots & \ddots \\ 1 & \ldots & \ldots & \ddots & \ddots \\ \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 & 1 & \ldots \\ 0 & 1 & 2 & 3 & \ldots \\ 0 & 0 & 1 & 3 & \ddots \\ 0 & 0 & 0 & \ldots & \ddots \\ 0 & \ldots & \ldots & \ddots & \ddots \\ \end{pmatrix}.$

Exercise (***). Give a proof of a Menger's theorem based on the results of the exercise above: Let G be a finite graph and p and q two vertices that are not neighbors. Then the size of the minimum vertex cut for p and q (the minimum number of vertices whose removal disconnects p and q) is equal to the maximum number of pairwise vertex-independent paths from p to q.