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Unformatted text preview: (d) Show that there are only two non-trivial proper subgroups of this group, and find them. (7) (Total 29 marks) 111. Consider a group (G, o) with identity e. Suppose that H is a subset of G such that H = {x ∈ G : x o a = a o x, for all a ∈ G}. Show that (H, o) is a subgroup of (G, o), by showing that (a) e ∈ H; (2) 76 (b) if x, y ∈ H, then x o y ∈ H, [i.e. show that (x o y) o a = a o (x o y)]; (5) (c) if x ∈ H, then x–1 ∈ H. (4) (Total 11 marks) 112. The network of cities in a certain region served by an airline is shown in the following diagram, with edges representing direct connections. A B (a) E C F D Copy and complete the table below which shows the least number of edges connecting each pair of cities in this network. (Such a table gives the least number of stops required between cities in this region.) (4) A B C D E F A B 1 C D E 2 F 77 (b) The accessibility index is used to determine how easy it is to get to a particular city. To calculate the accessibility index for a given city, the total of each column is divided by the degree of the vertex representing it. The most accessible city is the one with the smallest accessibility index, and the least accessible city has the largest accessibility index. (i) Which city in this region is the most accessible and which city is the least accessible? Give your reasons. (4) (ii) A new flight is added between cities A and C. With this change, which city is the most accessible and which city is the least accessible? Show your working. (3) (Total 11 marks) 113. Below are two graphs U and V with 8 vertices each. 8 A 1 7 D E F G H 3 5 4 U (a) C 2 6 B V Set up an adjacency matrix for graph U. (3) (b) By setting up an appropriate adjacency matrix for V, show that the two graphs are isomorphic. (6) (c) Determine whether V is planar. (4) (Total 13 marks) 78 114. In this part, marks will only be awarded if you show the correct application of the required algorithms, and show all your working. In an offshore drilling site for a large oil company, the distances between the planned wells are given below in metres. 1 2 3 4 5 6 7 8 9 2 40 60 4 90 190 130 5 80 200 10 160 6 70 40 20 40 130 7 60 120 50 90 30 60 8 50 140 90 70 140 70 40 9 40 170 140 60 50 90 50 70 10 200 80 150 110 90 30 190 90 100 11 (a) 30 3 150 30 200 120 190 120 60 190 150 10 200 It is intended to construct a network of paths to connect the different wells in a way that minimises the sum of the distances between them. Use Prim's algorithm to find a network of paths of minimum total length that can span the whole site. (8) (b) Pipes are laid under water. Well 1 has the largest amount of oil to be pumped per day, and Well 11 is designed to be the main transportation hub. The only possible connections to be made between wells are shown in the diagram below. 6 2 1 7 10 11 5 9 3 4 8 79 The associated cost for each pipe, in 100-thousand dollar figures, are given in the table below. Use Dijkstra's algorithm to find the path with minimum co...
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