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184a_01f_fe - Math 184A Final Exam 5 December 2001 Please...

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Math 184A Final Exam 5 December 2001 Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but you may have two pages of notes. Calculators are NOT allowed. You must show your work to receive credit. 1. In each case, give an example or explain why none exists . (a) A permutation f of { 1 , 2 , 3 , 4 , 5 } such that f 20 has no fixed points. (b) A simple graph with 8 vertices, 2 connected components and 12 edges. (c) A connected simple graph with 8 vertices, 8 edges and no cycles. 2. Let V = { 1 , 2 , . . . , n } . An oriented simple graph with vertex set V is a simple graph with vertex set V where each edge is given a direction. In other words, given two vertices v and w in V , there are three choices: (a) no edge between them, (b) an edge ( v, w ) or (c) an edge ( w, v ). (Unlike a directed graph, you cannot have both ( v, w ) and ( w, v ) as edges and you cannot have loops.) Let N = ( n 2 ) . We showed that there are 2 N simple graphs with vertex set V and ( N q ) of them exactly q edges.
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