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METHODS OF DISCRETE MATHEMATICS. Homework Problems Relevant to the Final Exam. Due: Thursday, January 13, ... a perfect score before totaling and averaging.
Typology: Schemes and Mind Maps
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Homework Problems Relevant to the Final Exam Due: Thursday, January 13, 2004
Required Problems Note: in calculating homework scores, we will implement a “throw out your worst set” policy by coveting your worst grade (in terms of missing points) to a perfect score before totaling and averaging. If you have done well on all other problem sets, you may lose little by not handing this in. However, you will need to know how to do the problems in order to be well prepared for the final exam!
(a) Show the order in which routes (edges) will be added to form the minimum spanning tree if you start in Washington and use Prim’s algorithm (the one described on Biggs section 16.3) (b) Show the order in which routes (edges) will be added to form the minimum spanning tree if you use Kruskal’s algorithm (described only in the notes, where during the construction of the spanning tree the selected edges may fail to form a single tree.)
Exploratory Problems Here is a very large set, so that you can have plenty of choice. Most of the problems cover material discussed at the last class, on January 13. You may hand in solutions to these problems at the final exam. If you are sure that you already have the maximum of 50 points from exploratory problems and programming projects, please do not hand them in! Do them, though, especially the final three, since they provide a nice review of symmetry groups for the final exam.
(a) Show that (rf )^3 = 1, and find a different element of order 5 for which rf does not have order 3. (b) The graph of the icosahedral group (Grossman and Magnus, figure 16.2) shows a central pentagon with five vertices of order 2. Ex- press these group elements as words in r and f (without using r−^1 ). Express them as permutations by using the specific r and f given above.
(d) Circle the vertices of the graph that correspond to the elements of the subgroup A 4.