Foundations of Data Science

a
c
R=1 R=2
R=1
b
R=2
d
R=1
Figure 5.14: An electrical network of resistors.
u v
u v u v
Figure 5.15: Three graphs
Exercise 5.27 (Thomson’s Principle) The energy dissipated by the resistance of edge xy
in an electrical
∑
network is given by i 2 xyr xy . The total energy dissipation in the network
is E = 1 i 2 2 xyr xy where the 1 accounts for the fact that the dissipation in each edge is
2
x,y
counted twice in the summation. Show that the actual current distribution is the distribution
satisfying Ohm’s law that minimizes energy dissipation.
Exercise 5.28 (Rayleigh’s law) Prove that reducing the value of a resistor in a network
cannot increase the effective resistance. Prove that increasing the value of a resistor cannot
decrease the effective resistance. You may use Thomson’s principle Exercise 5.27.
Exercise 5.29 What is the hitting time h uv for two adjacent vertices on a cycle of length
n? What is the hitting time if the edge (u, v) is removed?
Exercise 5.30 What is the hitting time h uv for the three graphs if Figure 5.15.
Exercise 5.31 Show that adding an edge can either increase or decrease hitting time by
calculating h 24 for the three graphs in Figure 5.16.
Exercise 5.32 Consider the n vertex connected graph shown in Figure 5.17 consisting
of an edge (u, v) plus a connected graph on n − 1 vertices and m edges. Prove that
h uv = 2m + 1 where m is the number of edges in the n − 1 vertex subgraph.
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