Science:Math Exam Resources/Courses/MATH307/April 2013/Question Section 201 05 (a)

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MATH307 April 2013

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Question Section 201 05 (a)
Consider the following graph, interpreted as a resistor network with all resistances R = 1. (a) Write down the incidence matrix D and the Laplacian matrix L for this graph. Show that (for any graph) N(L) = N(D). Is it true that R(DT) = R(L)? Give a reason.

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Solution

${\text{Write the incidence matrix }}D{\text{ and Laplacian matrix }}L{\text{. Also show that (for any graph) }}N(L)=N(D){\text{. Is it true that }}R(D^{T})=R(L){\text{?}}$ Answer: Reading off the graph (where -1 is info leaving node and +1 is info entering node): $D={\begin{bmatrix}-1&1&0&0&0&0\\0&-1&1&0&0&0\\-1&0&0&1&0&0\\0&-1&0&0&1&0\\0&0&-1&0&0&1\\0&0&0&-1&1&0\\0&0&0&0&-1&1\end{bmatrix}}$ and can either compute L by } L = D^TD \text{ or get it directly from the graph using the following method: $L_{i,i}={\text{number of other nodes connected to the }}i^{th}{\text{ node}}$ $L_{i,j}=-1{\text{if the two nodes are connected, }}0{\text{ otherwise}}$ $L={\begin{bmatrix}2&-1&0&-1&0&0\\-1&3&-1&0&-1&0\\0&-1&2&0&0&-1\\-1&0&0&2&-1&0\\0&-1&0&-1&3&-1\\0&0&-1&0&-1&2\\\end{bmatrix}}$ As for nullspaces of L and D, $L=D^{T}RD=D^{T}D$ ${\text{so if }}D{\underline {v}}={\underline {0}}$ $L{\underline {v}}=d^{T}D{\underline {v}}$ $=D^{T}({\underline {0}})$ $={\underline {0}}$ $\therefore N(L)=N(D)$ Also: $R(D^{T})=N(D)^{\perp }=N(L)^{\perp }{\text{ by orthogonality relations}}$ $=R(L^{T})$ $=R(L){\text{ since L is symmetric}}$ $\therefore R(D^{T})=R(L)$

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