184a_03f_2e

184a_03f_2e - a n = 2 n n-long sequences of zeroes and...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 184A Second Exam 24 November 2003 Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but you may have a page of notes. Calculators are NOT allowed. You must show your work to receive credit. 1. (16 pts.) An oriented graph is a simple graph in which each edge has been given a direction. In other words, given vertices x and y with x n = y , exactly one of the following is true: There is no edge between x and y . There is an edge from x to y . There is an edge from y to x . Obtain formulas for (a) the number of oriented graphs with vertex set n ; that is, the number of n -vertex oriented graphs; (b) the number of n -vertex oriented graphs having exactly k edges. 2. (a) (2 pts.) Sketch the simple graph G = ( V, E ) where V = { a, b, c, d, e } E = b { a, b } , { a, c } , { a, d } , { b, c } , { d, e } B . (b) (6 pts.) Compute the chromatic polynomial P G ( x ). (c) (2 pts.) How many ways can G be properly colored if 5 colors are available? 3. (24 pts.) There are
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a n = 2 n n-long sequences of zeroes and ones, including the empty sequence, and so A ( x ) = a n x n = (1 2 x )-1 . (You do not need to derive this.) Let f n be the number of such sequences that do not contain the pattern 11100. Let F ( x ) = f n x n . (a) Derive either of the two formulas A ( x ) = F ( x ) + A ( x ) x 5 F ( x ) A ( x ) = s t =0 ( F ( x ) x 5 ) t F ( x ) . (Both formulas are correct. Which you derive will depend on how you think about the problem.) (b) Using either of the formulas in (a) and the formula for A ( x ), Fnd polynomials P ( x ) and Q ( x ) so that F ( x ) = P ( x ) Q ( x ) ; for example, F ( x ) might be 7 23-x 9 . (c) Using (b) or otherwise, obtain a simple recursion for f n for n 5. Dont worry about initial conditions. Final Exam in Center 113 END O EXAM...
View Full Document

Ask a homework question - tutors are online