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 LG, 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 vk → vl means that there is a directed edge from vertex vk to the vertex vl, 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 LG 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} $$

The Schur complement of the matrix $$L_G$$ w/respect of the invertible block D is the matrix $$ L_G/D = A-BD^{-1}C.$$


 * Exercise (*). Prove the following determinant identity:$$\det L_G = \det(L_G/D) \det D.$$

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 vk occupies, is the boundary vertex vl. 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 pk and pl of a graph G

w_{kl} = \sum_{v_k\xrightarrow[]{path} v_l}\prod_{e=(p,q)\in path}w(e)/l_{pp}, $$ Hint: use the Leibniz definition of the determinant


 * for two distinct boundary vertices vk and vl 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}. $$