Science:Math Exam Resources/Courses/MATH307/April 2012/Question 04 (d)
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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.
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The formula for the Laplacian matrix L is:
But we can simply calculate the Laplacian matrix by computing the following:
For the diagonal entries ():
For the non-diagonal edges (): if there are shared edges and if no shared edges
Where R is the resistance of the edge. which in this question so .
Following the equations given above, the Laplacian matrix is:
The columns of matrix L are labeled as node 1 to 6 from left to right.