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CombinatoricsandGraphTheory2005Fall - Qualifying Exam Fall...

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Unformatted text preview: Qualifying Exam, Fall 2005 Graph Theory and Combinatorics 1. (a) Define a (b, v,r,k,)\)—design. (b) Derive the following identities: I l bk = or r(k— 1) = /\(v— 1) (c) What are the symmetric designs called for which k = 3 and /\ = 1? Derive necessary congruence conditions on v for such a design to exist. 2 (a) Prove Euler’s formula relating the numbers of vertices, edges and faces of a connected planar map. (Hint: induce on the number of edges.) (b) From this, derive the formula 1 1 1 1 -+-=-+—, r s e 2 where r denotes the average valence, 3 denotes the average covalence, and 6 denotes the number of edges. (c)-1Prove that every finite planar map has a face of covalence at most 5. (d) Prove that if a planar map has no face with covalence less than 5, then it has at least 12 faces of covalence exactly 5. Finally, conclude th this bound is sharp. ”I no \Sol VLP‘l" L25 3. (a) Prove that if a graph‘ifs edge- transitive but not vertex— transitive, then it is bipartite. (b) Prove that Cayley graphs are vertex— transitive but not necessarily edge-transitive. (c) Prove that connected, edge-transitive graphs have trivial atoms. 4. .Let X be a connected k- valent graph with diameter 6, where 19,6 > 2. (a) As a function of k and 6, what is the largest possible value of lV(X )|? (b) What is the girth of such a maximal graph? . (c ) For what values of k and 6 is this bound known to be attainable, and for what values is it known not to be attainable? (d) Is such a graph distance—regular, and if so, what is its distance array? .. 5. Let positive integers n, m and k be given with k < 71, let A1, . . . ,Am be disjoint n-sets and let E = UlilAi- Define I Q E to belong to I if }I flAl-I S k for all 2'. (a) Prove that I is the collection of independent sets for a matroid on E. (b) Describe the basfs and ran> function for this matroid and the cycles of its dual. 6. How many integer solutions are there to the equation $1+$2+$3+$4+$5=20 {3 where 1 3 mi 3 6, for 2' = 1, 2,3,4? [You need not execute all of the arithmetic. It suffices to indicate what calculations need to be performed] ...
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