# Suppose you are applying the algorithm we learned to find a Hamiltonian cycle. You are at a…

MATH 475 Final Exam Version 2 5/14 1. Consider the succession 0, 2, -6, 16, ... resulting from a homogeneous straight repose bearing of dispose 2. Find: (a) The expend constants u, v so that an = uan-1 + van-2. (b) A unavailable constructula for an. (c) The unavailable construct of the generating duty for an. 2. Let hn be the reckon of permutations of n of the lore a, b, c, d, e where #a = 0 or 2, #b, c = 0, #d =odd nonnegative and #e =even nonnegative. Transcribe the exponential generating duty for hn in: (a) A unavailable construct. (b) The construct h0 + h1x + h2x2 + 1Pn=3hn xn n! 3. For (a),(b) transcribe down the unavailable construct of the generating duty and for (c) transcribe down the unavailable construct of the exponential generating duty. (a) 0, 1, 12 , 13 , 14 , ... (b) 0,-1/3 0 ,1/3 1 ,-1/3 2 , ... (c) 1, 1 2! , 1 3! , 1 4! , ... 4. Suppose f(x) = ex sin x is the generating duty for a succession hn. First invent a constructula for f(n)(x) and use this to invent a unavailable constructula for hn. 5. Examine that the quality succession (d1, ..., dn) of a single-minded graph satisfies d1 + ... + dn = P(n, 2) and then bestow an illustration of a quality succession satisfying this which is not the quality succession of a single-minded graph. 6. Suppose you are applying the algorithm we versed to invent a Hamiltonian cycle. You are at a object where you accept a route which cannot be bland but has neighboring endpoints. There are calm?} vertices left to be comprised. Explain what the proximate march is and examine that it can be performed. 7. What criteria on each of the aftercited would fix that the graph has an Eulerian transfer? Check twain unavailable and known. Justify little. (a) Kn (b) Km,n (c) T a tree 8. (a) Transcribe down the Pr¨ufer succession selfsame to the tree shown here: 9 1 3 2 5 86 7 4 (b) Bestow an illustration of two labeled trees whose Pr¨ufer successions are rearrangements of one another but the trees themselves are not isomorphic when unlabled. (c) By looking at the Pr¨ufer succession for a labeled tree of dispose n, how can you report that the tree has correspondently two...

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