Science:Math Exam Resources/Courses/MATH307/April 2012/Question 04 (d)

MATH307 April 2012

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Question 04 (d)

Consider a network arranged in the shape of an octahedron as in this diagram: The nodes have been labeled with the large numbers and 3 of the edges (resistors) have been given orientations and labeled with small numbers. Assume all resistances are equal to 1. Let D be the incidence matrix (for some choice of labeling and arrows for the remaining edges) and let L be the Laplacian. (d) Write down the Laplacian L.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The formula for the Laplacian matrix L is: $L=D^{T}R^{-1}D$ But we can simply calculate the Laplacian matrix by computing the following: For the diagonal entries ($L_{ii}$): $L_{ii}=\sum _{connected\ edges,i}\gamma _{i}$ For the non-diagonal edges ($L_{ij}$): $L_{ij}=\sum _{shared\ edgesj}\gamma _{j}$ if there are shared edges and $L_{ij}=0$ if no shared edges $\gamma ={\frac {1}{R}}$ Where R is the resistance of the edge. which in this question $R=1$ so $\gamma =1$. Following the equations given above, the Laplacian matrix is: $L={\begin{bmatrix}4&-1&-1&-1&-1&0\\-1&4&-1&0&-1&-1\\-1&-1&4&-1&0&-1\\-1&0&-1&4&-1&-1\\-1&-1&0&-1&4&-1\\0&-1&-1&-1&-1&4\end{bmatrix}}$ The columns of matrix L are labeled as node 1 to 6 from left to right. FINAL ANSWER $L={\begin{bmatrix}4&-1&-1&-1&-1&0\\-1&4&-1&0&-1&-1\\-1&-1&4&-1&0&-1\\-1&0&-1&4&-1&-1\\-1&-1&0&-1&4&-1\\0&-1&-1&-1&-1&4\end{bmatrix}}$