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#2: Let G be the graph with vertex set Z_2^n( the same as the n-cube) and with edge set defined as follows: {u, v} is an edge of G if u and v differ in exactly two coordinates (i.e., if ω( u, v) = 2). What are the eigenvalues of G? #4: Let r, s ≥ 1. The complete bipartite graphK rs has vertices u1,u2...,ur , v 1, v 2, ... vs, with one edge between each u i and v j (so rs edges in all). (a) By purely combinatorial reasoning, compute the number of closed walks of length l in K rs . (b) Deduce from (a) the eigenvalues of K rs . #5: Let H n be the complete bipartite graph K nn with n vertex-disjoint edges removed. Thus H n has 2n vertices and n( n - 1) edges, each of degree (number of incident edges) n - 1. Show that the eigenvalues of G are ± 1 (n - 1 times each) and ± (n - 1) (once each). #9: Let G be a (finite) graph with vertices v1,...,vp and eigenvalues λ1,...λp . We know that for any i, j there are real numbers c1(i,j),...cp(i,j) such that for all l ≥ 1, (a) Show that ck(i, i) ≥ 0. (b) Show that if i ≠ j then we can have c k (i, j) < 0. (The simplest possible example will work.)
a piece of wire is 38 inches long. the wire is cut into two pieces; then each piece is bent to form a square. Find the length of each piece if three times the area of the larger square exceeds the area of the smaller square by 95.75 square inches.
The sample results were x= 357.5, s= 117.29. Logger working on the road claims the skidding distance is at least 400 meters. Is there sufficient evidence to refute this claim?
Probability that Lounging Larry wins the horse race. The odds against Lounging Larry winning the horse race are 5:12.
What is the area of the region bounded by y=sin(x), y=cos(x), and the x-axis between x=0 and x=pi/2
Suppose the farmer buys insurance for $3000 that pays him $20,000 in the event of a severe storm that destroys much of the crop, and pays nothing otherwise. Find the probability distribution of his profit . Find the mean, and summarize the effect ..
Describe the two methods to simplify a complex fraction. Which of the two methods for simplifying a complex fraction is simpler? Explain your reasoning.
At square deals floor covering a popular floor covering sells for $9.00 per square foot . Write a function that gives the price ( P) of a square floo covering that has edge length of (x) feet.
Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.
Satisfies the Cauchy-Riemann equations
A farmer has a rectangular garden plot surrounded by 200 ft of fence. Find the length and width of the garden if its area is 1275 ft2.
How is dividing a polynomial by a binomial similar to or different from the long division you learned in elementary school? Can understanding how to do one kind of division help you with understanding the other kind? What are some examples from re..
A retailer spent $48 to purchase a number of special mugs. Two of them were broken in the store, but by selling each of the remaining mugs for 43 above the original cost per mug, she made a total profit of $22. Construct a quadratic equation that ..
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