2. Say that for any pair of people, they either both know each other, or they arestrangers (neither knows each other). Prove that in any party of six people, there isa group of three that either all know each other, or all are strangers.(Hint: how can you model this problem using a graph?)

Answered: Image /qna-images/answer/bdcb9d69-fdfd-45ee-ad9b-7472e108920b.jpg

Answered: 2. Say that for any pair of people,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. Consider the following graphs G₁, G2, and G3 G₁ is the graph with vertex set {1, 2, 3, 4, X5, X6, 7, 8}, and edge set {1x2, X1X7, X1X8, X2X3, X3X4, X3 X5, X3X6, X3X7, X4X5, X5X6, X7X8}. • G₂ is the graph with vertex set (y₁, Y2, Y3, Y4, Y5, Y6, y7, ys} and adjacency matrix: Y1 Y2 Y3 Y4 Y5 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 1 000 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 27 28 Y6 Y7 Y8 • G3 is the graph with vertex set {21, 22, 23, 24, 25, 26, 27, 28} and incidence matrix: 2₁ 1 1 22 23 24 25 26 1 0 0000 0 00 0 0 1 0 0 000 00 0 1 0 1 1 0000 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 00000000011 (a) Exactly two of the three graphs are ismorphic. Which two are they? (b) Provide an isomorphism between the two graphs you picked in part (a).
How long can a path be in the complete bipartite graph k(m,n)?
A large bakery has many different products for sale. Suppose that 200 come in between 6am and 10am. Of the 200 customers 140 order donuts, 100 order cinnamon rolls, and 80 order both. Make a Venn Diagram to model the situation.
In Flatland there are four cities, Abbott, Banchoff, Charlie and Dimension. The distance from The distance from Abbott to Banchoff is 9 miles. The distance from Abbott to Charlie is 8 miles. The distance from Abbott to Dimension is 11 miles. The distance from Banchoff to Charlie is 15 miles. The distance from Banchoff to Dimension is 13 miles. The distance from Charlie to Dimension is 12 miles. Sketch a graph to represent the traveling salesman problem for this situation, and find the length of the shortest path. Remember to begin and end in the same city. The shortest path is miles.
. Draw the graph of the problem and comment the solution. Max z = 10x1 + 20x2 s.t. 2x1 + 6x2 ≤ 10 2x1 + 4x2 ≤ 16 4x2 ≤ 8 x1 ≥ 5 x1, x2 ≥ 0
31. In any undirected graph the sum of degrees of all the nodes A. must be even B. are twice the number of edges C. must be odd D. need not be even
3. Prove the following statement: If G is a graph with at least 2 vertices, then a(G) > 2 or w(G) 2 2.
In 2011 the United States federal government decided that all businesses must pay their employees a minimum wage of $7.25 per hour. However, state governments could decide that businesses in their states should pay their employees more than the $7.25 per hour. Below is a graph of hourly minimum wage requirements for five states. Use the information provided in the graph to decide which of the following statements is NOT TRUE! New York's hourly wage requirement is about the same as the federal minimum wage requirement. The hourly wage in Ohio is more than $1 above the hourly wage in Minnesota. Oregon has the highest hourly wage requirement at over $8.00 per hour. The hourly wage in New York is more than double the hourly wage in Georgia.

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2. Say that for any pair of people, they either both know each other, or they are strangers (neither knows each other). Prove that in any party of six people, there is a group of three that either all know each other, or all are strangers. (Hint: how can you model this problem using a graph?)