Explanation Algorithm used: Prim’s algorithm for minimal spanning tree. Step 1 (Select starting vertex) Select vertex U . and set L ( U ) and T = ϕ Step 2 (enlarge T ) While some edge has one vertex in L and one not in L . (a) Choose an edge of greatest weight from among those with one vertex in L and the other not in L . (Ties can be broken arbitrarily.) (b) Place the edge in T (c) Place its vertices in L (one of these is already in L ). Endwhile Step 3 (is there a minimal spanning tree?) If | L | < n The graph is not connected, and so it has no minimal spanning tree. Otherwise The edges in T and their incident vertices from a minimal spanning tree. Prim’s algorithm for maximum spanning tree. Step 1 (Select starting vertex) Select vertex U . and set L ( U ) and T = ϕ Step 2 (enlarge T ) While some edge has one vertex in L and one not in L . Choose an edge of greatest weight from among those with one vertex in L and the other not in L . (Ties can be broken arbitrarily.) Place the edge in T Place its vertices in L (one of these is already in L ). Endwhile Step 3 (is there a maximal spanning tree?) If | L | < n The graph is not connected, and so it has no maximal spanning tree. Otherwise The edges in T and their incident vertices from a maximal spanning tree. Spanning tree. A snapping tree of a graph G is a tree formed by containing all the vertices of the original graph G . Graph: Consider the following graph for an example. The minimal spanning tree can be obtained as below. Begin with vertex A , set L = { A } and T = ϕ . There are edges that one vertex in the set L and other vertices which are not in L . The edges on A that do not have their other vertices in set L are a, b and d and a has the smallest weight. Therefore, include the vertex B is in L . Therefore, set L = { A , B } and T = { a } . The edges which have exactly one vertex in the set L are b, c, d and e . The edge c has the smallest weight among these vertices. Therefore, include c in T and vertex E in L . Update the sets as L = { A , B , E } and T = { a , c } . Consider the edges b, d, e and h . The edge g has the smallest weight. Therefore, include C in L and g in T . Update the sets as L = { A , B , E , C } and T = { a , c , g }

5th Edition

ISBN: 9780134689562

Author: Dossey, John A.

Publisher: Pearson,

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Chapter 5.2, Problem 31E

To determine

To graph: An example of a connected weighted graph in which the same edge is part of every minimal spanning tree and every maximal spanning tree.

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Chapter 5.2, Problem 30E

Chapter 5.2, Problem 32E

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A function is defined on the interval (-π/2,π/2) by this multipart rule: if -π/2 < x < 0 f(x) = a if x=0 31-tan x +31-cot x if 0 < x < π/2 Here, a and b are constants. Find a and b so that the function f(x) is continuous at x=0. a= b= 3

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = (x + 4x4) 5, a = -1 lim f(x) X--1 = lim x+4x X--1 lim X-1 4 x+4x 5 ))" 5 )) by the power law by the sum law lim (x) + lim X--1 4 4x X-1 -(0,00+( Find f(-1). f(-1)=243 lim (x) + -1 +4 35 4 ([ ) lim (x4) 5 x-1 Thus, by the definition of continuity, f is continuous at a = -1. by the multiple constant law by the direct substitution property

Chapter 5 Solutions

Discrete Mathematics

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Ch. 5.1 - Seven farming communities in Iowa want to develop...Ch. 5.1 - As few trails as possible are to be built between...Ch. 5.1 - A farmer needs to irrigate the fields in which his...Ch. 5.1 - Draw a graph that is not a tree for which the...Ch. 5.1 - Prob. 15E Ch. 5.1 - What is the smallest number of edges in a...Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - There are two saturated hydrocarbons of the type...Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - In order to count the number of distinct trees...Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - In order to count the number of distinct trees...Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - 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