Explanation Algorithm used: Dijkstra’s algorithm. The weight of the directed edges on vertices U and V is given by W ( U , V ) . If there doesn’t exist an edge which has a direction on U and V then it is written that W ( U , V ) = ∞ . Being with labelling S . Assign the label S as 0. Set that P = { S } . Now label the vertices. Assign each vertex V which is not in P with the label W ( S , V ) and set V has the predecessor S . (Enlarge P and revise labels.) Repeat the process. (Make another label permanent.) Inclusive of P a vertex U which have the smallest label of the vertices which is not in P . Now revise the temporary labels. For each vertex X which is not in P that is adjacent to U , replace the label on X with the smaller of the old label which is on X and the sum of the label on U and W ( U , X ) . If the label on the X is changed. Consider that U is the new predecessor of X . Until and unless P contains every vertex of the graph G . (Find the distances and shortest directed paths.) The label on a vertex y indicates its direction for S . Description: Consider the given figure. Step 1: Begin with setting P = { S } and assign label 0 to S . Step 2: Assign the label W ( S , V ) and the predecessor S to every other vertex V. Therefore, W ( S , V ) = ∞ . This is in case there is no edge which joins S and V. Step 3: The vertex which is not in P with the smallest label is F

5th Edition

ISBN: 9780134689562

Author: Dossey, John A.

Publisher: Pearson,

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Chapter 4.5, Problem 67E

To determine

The distance from S to all the other vertices in the directed weighted graph. Also, find the shortest directed path from S to A for the given graph.

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Chapter 4.5, Problem 66E

Chapter 4.5, Problem 68E

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(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

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Discrete Mathematics

Ch. 4.1 - In Exercises 1–4, list the set of edges and set of...Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - In Exercises 5–8, draw a diagram representing the...Ch. 4.1 - Prob. 7E Ch. 4.1 - In Exercises 5–8, draw a diagram representing the...Ch. 4.1 - Prob. 9E Ch. 4.1 - In Exercises 9–14, determine whether a graph is...

Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Draw the graph with ν = {1, 2, … , 10} as its set...Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Show that there are an even number of vertices...Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - In Exercises 26–29, find the adjacency matrix and...Ch. 4.1 - Prob. 29E Ch. 4.1 - In Exercises 30 and 31, construct the graph for...Ch. 4.1 - Prob. 31E Ch. 4.1 - In Exercises 32 and 33, construct the graph for...Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - In Exercises 35-37, can each matrix be an...Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Are the pairs of graphs in (a), (b), and (c)...Ch. 4.1 - Are the pairs of graphs in (a), (b), and (c)...Ch. 4.1 - Draw all the non isomorphic graphs with three...Ch. 4.1 - Prob. 45E Ch. 4.1 - Draw all the nonisomorphic graphs with five...Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Suppose a graph has n vertices, each with degree...Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Suppose Mr. and Mrs. Lewis attended a bridge party...Ch. 4.1 - Prove that if a graph has at least two vertices,...Ch. 4.2 - In Exercises 1–4, determine whether the multigraph...Ch. 4.2 - Prob. 2E Ch. 4.2 - Prob. 3E Ch. 4.2 - Prob. 4E Ch. 4.2 - Prob. 5E Ch. 4.2 - Prob. 6E Ch. 4.2 - Prob. 7E Ch. 4.2 - Prob. 8E Ch. 4.2 - Prob. 9E Ch. 4.2 - In Exercises 9 and 10, perform the following...Ch. 4.2 - Prob. 11E Ch. 4.2 - Prob. 12E Ch. 4.2 - Prob. 13E Ch. 4.2 - Prob. 14E Ch. 4.2 - Prob. 15E Ch. 4.2 - Prob. 16E Ch. 4.2 - Prob. 17E Ch. 4.2 - In Exercises 18–23, determine whether the...Ch. 4.2 - Prob. 19E Ch. 4.2 - In Exercises 18–23, determine whether the...Ch. 4.2 - Prob. 21E Ch. 4.2 - In Exercises 18–23, determine whether the...Ch. 4.2 - Prob. 23E Ch. 4.2 - In Exercises 24–29, determine whether the...Ch. 4.2 - Prob. 25E Ch. 4.2 - Prob. 26E Ch. 4.2 - Prob. 27E Ch. 4.2 - Prob. 28E Ch. 4.2 - Prob. 29E Ch. 4.2 - Prob. 30E Ch. 4.2 - Prob. 31E Ch. 4.2 - Prob. 32E Ch. 4.2 - Prob. 33E Ch. 4.2 - Prob. 34E Ch. 4.2 - Prob. 35E Ch. 4.2 - Prob. 36E Ch. 4.2 - Prob. 37E Ch. 4.2 - An old childhood game asks children to trace a...Ch. 4.2 - Prob. 39E Ch. 4.2 - In 1859, Sir William Rowan Hamilton, a famous...Ch. 4.2 - Give examples of connected graphs satisfying each...Ch. 4.2 - Prob. 42E Ch. 4.2 - Prob. 43E Ch. 4.2 - Prob. 44E Ch. 4.2 - Prob. 45E Ch. 4.2 - Prob. 46E Ch. 4.2 - Prob. 47E Ch. 4.2 - Prob. 48E Ch. 4.2 - Prob. 49E Ch. 4.2 - Are the following two graphs isomorphic? Justify...Ch. 4.2 - Prob. 51E Ch. 4.2 - A bipartite graph is a graph in which the vertices...Ch. 4.2 - Prob. 53E Ch. 4.2 - Prob. 54E Ch. 4.2 - Prob. 55E Ch. 4.2 - Prob. 56E Ch. 4.2 - Prob. 57E Ch. 4.2 - Prob. 58E Ch. 4.2 - Prob. 59E Ch. 4.2 - Prob. 60E Ch. 4.2 - Prob. 61E Ch. 4.2 - Prob. 62E Ch. 4.2 - Prob. 64E Ch. 4.2 - Prob. 65E Ch. 4.3 - In Exercises 1–4, use the breadth-first search...Ch. 4.3 - In Exercises 1–4, use the breadth-first search...Ch. 4.3 - Prob. 3E Ch. 4.3 - In Exercises 1–4, use the breadth-first search...Ch. 4.3 - In Exercises 5–8, determine the distance from S to...Ch. 4.3 - In Exercises 5–8, determine the distance from S to...Ch. 4.3 - Prob. 7E Ch. 4.3 - Prob. 8E Ch. 4.3 - Prob. 9E Ch. 4.3 - Prob. 10E Ch. 4.3 - Prob. 11E Ch. 4.3 - Prob. 12E Ch. 4.3 - Prob. 13E Ch. 4.3 - Prob. 14E Ch. 4.3 - For the following graph, determine the number of...Ch. 4.3 - Prob. 16E Ch. 4.3 - Prob. 17E Ch. 4.3 - Prob. 18E Ch. 4.3 - Prob. 19E Ch. 4.3 - Prob. 20E Ch. 4.3 - Prob. 21E Ch. 4.3 - Prob. 22E Ch. 4.3 - Prob. 23E Ch. 4.4 - In Exercises 1–8, find the chromatic number of the...Ch. 4.4 - Prob. 2E Ch. 4.4 - Prob. 3E Ch. 4.4 - Prob. 4E Ch. 4.4 - Prob. 5E Ch. 4.4 - Prob. 6E Ch. 4.4 - Prob. 7E Ch. 4.4 - Prob. 8E Ch. 4.4 - Prob. 9E Ch. 4.4 - Prob. 10E Ch. 4.4 - Prob. 11E Ch. 4.4 - It might be supposed that if a graph has a large...Ch. 4.4 - Prob. 13E Ch. 4.4 - Prob. 15E Ch. 4.4 - Prob. 16E Ch. 4.4 - Prob. 17E Ch. 4.4 - Prob. 18E Ch. 4.4 - Prob. 19E Ch. 4.4 - Suppose is a graph with three vertices. How many...Ch. 4.4 - Prob. 21E Ch. 4.4 - Prob. 22E Ch. 4.4 - Prob. 23E Ch. 4.4 - Prob. 24E Ch. 4.4 - Prob. 25E Ch. 4.4 - Prob. 26E Ch. 4.4 - Prob. 27E Ch. 4.4 - Prob. 28E Ch. 4.4 - Prob. 29E Ch. 4.4 - Prob. 30E Ch. 4.4 - Prob. 31E Ch. 4.4 - Prob. 32E Ch. 4.4 - Prob. 33E Ch. 4.4 - Show that it is possible to assign one of the...Ch. 4.4 - Prob. 35E Ch. 4.4 - Prove Theorem 4.9 by mathematical induction on the...Ch. 4.4 - Suppose that each vertex of a graph is such that...Ch. 4.5 - In Exercises 1–4, list the vertices and directed...Ch. 4.5 - Prob. 2E Ch. 4.5 - Prob. 3E Ch. 4.5 - Prob. 4E Ch. 4.5 - Prob. 5E Ch. 4.5 - Prob. 6E Ch. 4.5 - Prob. 7E Ch. 4.5 - Prob. 8E Ch. 4.5 - Prob. 9E Ch. 4.5 - Prob. 10E Ch. 4.5 - Prob. 11E Ch. 4.5 - Prob. 12E Ch. 4.5 - Prob. 13E Ch. 4.5 - Prob. 14E Ch. 4.5 - Prob. 15E Ch. 4.5 - Prob. 16E Ch. 4.5 - Prob. 17E Ch. 4.5 - Prob. 18E Ch. 4.5 - Prob. 19E Ch. 4.5 - Prob. 20E Ch. 4.5 - Prob. 21E Ch. 4.5 - Prob. 22E Ch. 4.5 - Prob. 23E Ch. 4.5 - Prob. 24E Ch. 4.5 - Prob. 25E Ch. 4.5 - Prob. 26E Ch. 4.5 - Prob. 27E Ch. 4.5 - Prob. 28E Ch. 4.5 - Prob. 29E Ch. 4.5 - Prob. 30E Ch. 4.5 - Prob. 31E Ch. 4.5 - Prob. 32E Ch. 4.5 - Prob. 33E Ch. 4.5 - Prob. 34E Ch. 4.5 - Prob. 35E Ch. 4.5 - Prob. 36E Ch. 4.5 - Prob. 37E Ch. 4.5 - Prob. 39E Ch. 4.5 - Prob. 40E Ch. 4.5 - Prob. 41E Ch. 4.5 - Prob. 42E Ch. 4.5 - Prob. 43E Ch. 4.5 - Prob. 44E Ch. 4.5 - Prob. 45E Ch. 4.5 - Prob. 46E Ch. 4.5 - Prob. 47E Ch. 4.5 - Prob. 48E Ch. 4.5 - Prob. 49E Ch. 4.5 - Prob. 51E Ch. 4.5 - Prob. 52E Ch. 4.5 - Prob. 53E Ch. 4.5 - Prob. 54E Ch. 4.5 - Prob. 55E Ch. 4.5 - Prob. 56E Ch. 4.5 - Prob. 57E Ch. 4.5 - Prob. 58E Ch. 4.5 - Prob. 59E Ch. 4.5 - Prob. 60E Ch. 4.5 - Write a breadth-first search algorithm for...Ch. 4.5 - Prob. 62E Ch. 4.5 - Prob. 63E Ch. 4.5 - Prob. 64E Ch. 4.5 - Prob. 65E Ch. 4.5 - Prob. 66E Ch. 4.5 - Prob. 67E Ch. 4.5 - In Exercises 67–70, determine the distance from S...Ch. 4.5 - In Exercises 67–70, determine the distance from S...Ch. 4.5 - Prob. 70E Ch. 4.5 - Prob. 71E Ch. 4.5 - Prob. 72E Ch. 4.5 - Prob. 73E Ch. 4.5 - Prob. 74E Ch. 4.5 - Prob. 75E Ch. 4.5 - Determine whether the following pairs of directed...Ch. 4.5 - Determine whether the following pairs of directed...Ch. 4.5 - Prob. 78E Ch. 4.5 - Prob. 79E Ch. 4.5 - Prob. 80E Ch. 4.5 - Prob. 82E Ch. 4.5 - Prob. 83E Ch. 4.5 - Prob. 84E Ch. 4 - Prob. 1SE Ch. 4 - Prob. 2SE Ch. 4 - Prob. 3SE Ch. 4 - Prob. 4SE Ch. 4 - Prob. 5SE Ch. 4 - Prob. 6SE Ch. 4 - Prob. 7SE Ch. 4 - Prob. 8SE Ch. 4 - Prob. 9SE Ch. 4 - Prob. 10SE Ch. 4 - Prob. 11SE Ch. 4 - Prob. 12SE Ch. 4 - Prob. 13SE Ch. 4 - Prob. 14SE Ch. 4 - Is the property “is connected” a graph isomorphism...Ch. 4 - Prob. 16SE Ch. 4 - Prob. 17SE Ch. 4 - Prob. 18SE Ch. 4 - Prob. 19SE Ch. 4 - Prob. 20SE Ch. 4 - Prob. 21SE Ch. 4 - Prob. 22SE Ch. 4 - Prob. 23SE Ch. 4 - Prob. 24SE Ch. 4 - Prob. 25SE Ch. 4 - Prob. 26SE Ch. 4 - Prob. 28SE Ch. 4 - Prob. 29SE Ch. 4 - Prob. 30SE Ch. 4 - Prob. 31SE Ch. 4 - Prob. 32SE Ch. 4 - Prob. 34SE Ch. 4 - Prob. 35SE Ch. 4 - Prob. 36SE Ch. 4 - Prob. 37SE Ch. 4 - Prob. 38SE Ch. 4 - Prob. 39SE Ch. 4 - Prob. 40SE Ch. 4 - Prob. 6CP Ch. 4 - Prob. 9CP Ch. 4 - Prob. 14CP

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