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Unformatted text preview: Combinatorics Qualifying Exam Practice Questions July 21, 2008 1. (a) Define “( v, k, λ )-design”. Derive some identities involving these three parameters. (b) How are these designs related to finite projective planes? Give a picture of the simplest finite projective plane. (c) How are these related to Hadamard matrices? Give three of the simplest Hadamard matrices. (d) What values of n are possible for an n × n Hadamard matrix to exist? Prove this. Does an n × n Hadamard matrix indeed exist for all of these values? 2. Derive a recurrence relation for the number of ternary sequences of length n having no consecutive zeroes. Solve this recurrence relation by means of a generating function. 3. (a) Prove that K 6 is embeddable in the projective plane but K 7 is not. (b) Identify the dual of K 6 in the projective plane. (c) Identify the dual of K 7 in the torus. [Hint: Consider its girth and diameter.] 3. (a) Prove that if a finite directed graph is vertex-transitive, then the in-valence at each vertex equals its out-valence....
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This note was uploaded on 06/19/2011 for the course MATH 680 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.
- Spring '11