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

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. (c) Is is true that N(D) = N(L)? Give a reason. 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2012/Question 04 (c)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Yes, it is true that the nullspace of the incidence matrix (D) is equal to the nullspace of the Laplacian matrix. The reason for this is that the Laplacian matrix represents the total current flowing into or out of a node: According to Kirchoff’s Current Law, current flowing into a node must equal to the current flowing out of the node, there cannot be a buildup of current at any node if there are no batteries attached, which there isn’t in this question. This can be expressed as: For this to be true, because current flow has to be 0. For current flow to be equal to 0, you would have to have a voltage difference of 0, meaning that the voltage at each node is exactly the same, Therefore: 