Discrete Mathematics and Its Applications
8th Edition
ISBN: 9781260501759
Author: ROSEN
Publisher: MCG
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Textbook Question
Chapter 10.3, Problem 36E
Find an adjacency matrix for each of these graphs.
a)Kn
b) Cn
c) Wn
d)Km,n e)Qn
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Chapter 10 Solutions
Discrete Mathematics and Its Applications
Ch. 10.1 - Draw graph models, stating the type of graph...Ch. 10.1 - Prob. 2ECh. 10.1 - For Exercises 3-5, determine whether the graph...Ch. 10.1 - For Exercises 3-5, determine whether the graph...Ch. 10.1 - For Exercises 3-5, determine whether the graph...Ch. 10.1 - For Exercises 3-5, determine whether the graph...Ch. 10.1 - For Exercises 3-5, determine whether the graph...Ch. 10.1 - For Exercises 3-5, determine whether the graph...Ch. 10.1 - For Exercises 3-5, determine whether the graph...Ch. 10.1 - For each undirected graph in Exercises 3-9 that is...
Ch. 10.1 - Let G be a simple graph. Show that the relation R...Ch. 10.1 - Let G be an undirected graph with a loop at every...Ch. 10.1 - The intersection graphof a collection of...Ch. 10.1 - Use the niche overlap graph inFigure 11to...Ch. 10.1 - Construct a niche overlap graph for six species of...Ch. 10.1 - Draw the acquaintanceship graph that represents...Ch. 10.1 - Prob. 17ECh. 10.1 - Who can influence Fred and whom can Fred influence...Ch. 10.1 - Construct an influence graph for the board members...Ch. 10.1 - The word apple can refer to a plant, a food, or a...Ch. 10.1 - Prob. 21ECh. 10.1 - Which other teams did Team 4 beat and which teams...Ch. 10.1 - In a round-robin tournament the Tigers beat the...Ch. 10.1 - Construct the call graph for a set of seven...Ch. 10.1 - Explain how the two telephone call graphs for...Ch. 10.1 - a) Explain how graphs can be used to model...Ch. 10.1 - How can a graph that models e-mail messages sent...Ch. 10.1 - How can a graph that models e-mail messages sent...Ch. 10.1 - Describe a graph model that represents whether...Ch. 10.1 - Describe a graph model that represents a subway...Ch. 10.1 - Prob. 31ECh. 10.1 - Describe a graph model that represents the...Ch. 10.1 - Describe a graph model that represents traditional...Ch. 10.1 - Prob. 34ECh. 10.1 - Construct a precedence graph for the following...Ch. 10.1 - Describe a discrete structure based on a graph...Ch. 10.1 - Describe a discrete structure based on a graph...Ch. 10.1 - Prob. 38ECh. 10.2 - In Exercises 1-3 find the number of vertices, the...Ch. 10.2 - In Exercises 1-3 find the number of vertices, the...Ch. 10.2 - Prob. 3ECh. 10.2 - Prob. 4ECh. 10.2 - Can a simple graph exist with 15 vertices each of...Ch. 10.2 - Show that the sum, over the set of people at a...Ch. 10.2 - Prob. 7ECh. 10.2 - Prob. 8ECh. 10.2 - Prob. 9ECh. 10.2 - For each of the graphs in Exercises 7-9 determine...Ch. 10.2 - Construct the underlying undirected graph for the...Ch. 10.2 - What does the degree of a vertex represent in the...Ch. 10.2 - Prob. 13ECh. 10.2 - What does the degree of a vertex in the Hollywood...Ch. 10.2 - What do the in-degree and the out-degree of a...Ch. 10.2 - Prob. 16ECh. 10.2 - Prob. 17ECh. 10.2 - Show that in a simple graph with at least two...Ch. 10.2 - Use Exercise 18 to show that in a group of people,...Ch. 10.2 - Prob. 20ECh. 10.2 - In Exercises 21-25 determine whether the graph is...Ch. 10.2 - In Exercises 21-25 determine whether the graph is...Ch. 10.2 - Prob. 23ECh. 10.2 - Prob. 24ECh. 10.2 - In Exercises 21-25 determine whether the graph is...Ch. 10.2 - For which values ofnare these graphs bipartite?...Ch. 10.2 - Suppose that therearefour employees in the...Ch. 10.2 - Suppose that a new company has five employees:...Ch. 10.2 - Suppose that therearefive young women and five...Ch. 10.2 - Suppose that therearefive young women and six...Ch. 10.2 - Prob. 31ECh. 10.2 - Each of Exercises 31-33 can be solved using Hall's...Ch. 10.2 - Prob. 33ECh. 10.2 - Prob. 34ECh. 10.2 - Each of Exercises 31-33 can be solved using Hall's...Ch. 10.2 - Prob. 36ECh. 10.2 - How many vertices and how many edges do these...Ch. 10.2 - Prob. 38ECh. 10.2 - Prob. 39ECh. 10.2 - Prob. 40ECh. 10.2 - Prob. 41ECh. 10.2 - How many edges does a graph have if its degree...Ch. 10.2 - Prob. 43ECh. 10.2 - Determine whether each of these sequences is...Ch. 10.2 - Prob. 45ECh. 10.2 - Prob. 46ECh. 10.2 - Prob. 47ECh. 10.2 - Prob. 48ECh. 10.2 - Prob. 49ECh. 10.2 - Prob. 50ECh. 10.2 - Prob. 51ECh. 10.2 - Prob. 52ECh. 10.2 - Draw all sub graphs of this graph.Ch. 10.2 - Let G be a graph with vertices and e edges. Let M...Ch. 10.2 - For which values ofnare these graphs regular? a)...Ch. 10.2 - Prob. 56ECh. 10.2 - Prob. 57ECh. 10.2 - In Exercises 58-60 find the union of the given...Ch. 10.2 - Prob. 59ECh. 10.2 - In Exercises 58-60 find the union of the given...Ch. 10.2 - The complementarygraphGof a simple graph G has the...Ch. 10.2 - IfGis a simple graph with 15 edges andGhas 13...Ch. 10.2 - Prob. 63ECh. 10.2 - Prob. 64ECh. 10.2 - Prob. 65ECh. 10.2 - Prob. 66ECh. 10.2 - Prob. 67ECh. 10.2 - Describe an algorithm to decide whether a graph is...Ch. 10.2 - Theconverseof a directed graph G = (V, E), denoted...Ch. 10.2 - Theconverseof a directed graph G = (V, E), denoted...Ch. 10.2 - Prob. 71ECh. 10.2 - Prob. 72ECh. 10.2 - Theconverseof a directed graph G = (V, E), denoted...Ch. 10.2 - Prob. 74ECh. 10.2 - Theconverseof a directed graph G = (V, E), denoted...Ch. 10.3 - In Exercises 1-4 use an adjacency list to...Ch. 10.3 - Prob. 2ECh. 10.3 - Prob. 3ECh. 10.3 - Prob. 4ECh. 10.3 - Represent the graph in Exercise 1 with an...Ch. 10.3 - Represent the graph in Exercise 2 with an...Ch. 10.3 - Represent the graph in Exercise 3 with an...Ch. 10.3 - Represent the graph in Exercise 4 with an...Ch. 10.3 - Represent each of these graphs with an adjacency...Ch. 10.3 - In Exercises 10-12 draw a graph with the given...Ch. 10.3 - In Exercises 10-12 draw a graph with the given...Ch. 10.3 - In Exercises 10-12 draw a graph with the given...Ch. 10.3 - In Exercises 13-15 represent the given graph using...Ch. 10.3 - In Exercises 13-15 represent the given graph using...Ch. 10.3 - In Exercises 13-15 represent the given graph using...Ch. 10.3 - In Exercises 16-18 draw an undirected graph...Ch. 10.3 - In Exercises 16-18 draw an undirected graph...Ch. 10.3 - In Exercises 16-18 draw an undirected graph...Ch. 10.3 - Prob. 19ECh. 10.3 - In Exercises 19-21 find the adjacency matrix of...Ch. 10.3 - In Exercises 19-21 find the adjacency matrix of...Ch. 10.3 - In Exercises 22-24 draw the graph represented by...Ch. 10.3 - In Exercises 22-24 draw the graph represented by...Ch. 10.3 - In Exercises22-24 draw the graph represented by...Ch. 10.3 - Find the density of the graph in a)Figure...Ch. 10.3 - Prob. 26ECh. 10.3 - Prob. 27ECh. 10.3 - Prob. 28ECh. 10.3 - Is every zero-one square matrix that is symmetric...Ch. 10.3 - Prob. 30ECh. 10.3 - Prob. 31ECh. 10.3 - Prob. 32ECh. 10.3 - What is me sum of me entries in a column of me...Ch. 10.3 - What is the sum of the entries in a row of the...Ch. 10.3 - What is the sum of the entries in a column of the...Ch. 10.3 - Find an adjacency matrix for each of these graphs....Ch. 10.3 - Prob. 37ECh. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - In Exercises 38-48 determine whether the given...Ch. 10.3 - Show that isomorphism of simple graphs is an...Ch. 10.3 - Prob. 50ECh. 10.3 - Prob. 51ECh. 10.3 - Prob. 52ECh. 10.3 - Prob. 53ECh. 10.3 - Prob. 54ECh. 10.3 - Prob. 55ECh. 10.3 - Prob. 56ECh. 10.3 - Prob. 57ECh. 10.3 - How many non isomorphic simple graphs are there...Ch. 10.3 - How many nonisomorphic simple graphs are there...Ch. 10.3 - How many nonisomorphic simple graphs are there...Ch. 10.3 - Prob. 61ECh. 10.3 - Prob. 62ECh. 10.3 - Are the simple graphswiththe following adjacency...Ch. 10.3 - Determine whether the graphs without loops with...Ch. 10.3 - Prob. 65ECh. 10.3 - Prob. 66ECh. 10.3 - Prob. 67ECh. 10.3 - In Exercises 67-70 determine whether the given...Ch. 10.3 - Prob. 69ECh. 10.3 - In Exercises 67-70 determine whether the given...Ch. 10.3 - Show that ifGand H are isomorphic directed graphs,...Ch. 10.3 - Show that the property that a graph is bipartite...Ch. 10.3 - Prob. 73ECh. 10.3 - Prob. 74ECh. 10.3 - Prob. 75ECh. 10.3 - How much storage is needed to represent a simple...Ch. 10.3 - A devil's pairfor a purported isomorphism testis a...Ch. 10.3 - Prob. 78ECh. 10.4 - Does each of these lists of vertices form a path...Ch. 10.4 - Does each of these lists of vertices form a path...Ch. 10.4 - In Exercises 3-5 determine whether the given graph...Ch. 10.4 - In Exercises 3-5 determine whether the given graph...Ch. 10.4 - In Exercises 3-5 determine whether the given graph...Ch. 10.4 - How many connected components does each of the...Ch. 10.4 - What do the connected components of...Ch. 10.4 - Prob. 8ECh. 10.4 - Explain why in the collaboration graph of...Ch. 10.4 - In the Hollywood graph (see Example 3 inSection...Ch. 10.4 - Determine whether each of these graphs is strongly...Ch. 10.4 - Determine whether each of these graphs is strongly...Ch. 10.4 - What do the strongly connected components of a...Ch. 10.4 - Find the strongly connected components of each of...Ch. 10.4 - Find the strongly connected components of each of...Ch. 10.4 - Suppose that G=(V, E) is a directed graph. A...Ch. 10.4 - Prob. 17ECh. 10.4 - Prob. 18ECh. 10.4 - Find the number of paths of length n between two...Ch. 10.4 - Use paths either to show that these graphs are not...Ch. 10.4 - Prob. 21ECh. 10.4 - Prob. 22ECh. 10.4 - Prob. 23ECh. 10.4 - Find the number of paths of length n between any...Ch. 10.4 - Find the number of paths of length n between any...Ch. 10.4 - Find the number of paths between c andd inthe...Ch. 10.4 - Prob. 27ECh. 10.4 - Prob. 28ECh. 10.4 - Prob. 29ECh. 10.4 - Show that in every simple graph there is a path...Ch. 10.4 - In Exercises 31-33 find all the cut vertices of...Ch. 10.4 - In Exercises 31-33 find all the cut vertices of...Ch. 10.4 - Prob. 33ECh. 10.4 - Find all the cut edges in the graph sin Exercises...Ch. 10.4 - Prob. 35ECh. 10.4 - Prob. 36ECh. 10.4 - Prob. 37ECh. 10.4 - Prob. 38ECh. 10.4 - Prob. 39ECh. 10.4 - A vertex basis in a directed graph G is aminimal...Ch. 10.4 - Prob. 41ECh. 10.4 - Prob. 42ECh. 10.4 - Prob. 43ECh. 10.4 - Use Exercise43 to show that a simple graph with n...Ch. 10.4 - Show that a simple graph G withnvertices is...Ch. 10.4 - Prob. 46ECh. 10.4 - How many nonisom orphic connected simple graphs...Ch. 10.4 - Show that each of the following graphs has no cut...Ch. 10.4 - Prob. 49ECh. 10.4 - For each of these graphs, find(G),(G),and...Ch. 10.4 - Show that if G is a connected graph, then it is...Ch. 10.4 - Show that if G is a connected graph withnvertices...Ch. 10.4 - Find(Km,n) and(Km,n), wherem andnare positive...Ch. 10.4 - Construct a graphG with(G) - 1,(G) -2, and...Ch. 10.4 - Show that if G is a graph, then(G) (G).Ch. 10.4 - ExplainhowTheorem 2canbe used to find the length...Ch. 10.4 - Prob. 57ECh. 10.4 - Prob. 58ECh. 10.4 - Prob. 59ECh. 10.4 - Show that the existence of a simple circuit of...Ch. 10.4 - Prob. 61ECh. 10.4 - Use Exercise 61 to show that the...Ch. 10.4 - Prob. 63ECh. 10.4 - In an old puzzle attributed to Alcuin of York...Ch. 10.4 - Use a graph model and a path in your graph, as in...Ch. 10.4 - Prob. 66ECh. 10.5 - In Exercises 1-8 determine whether the given graph...Ch. 10.5 - In Exercises 1-8 determine whether the given graph...Ch. 10.5 - Prob. 3ECh. 10.5 - In Exercises 1-8 determine whether the given graph...Ch. 10.5 - In Exercises 1-8 determine whether the given graph...Ch. 10.5 - In Exercises 1-8 determine whether the given graph...Ch. 10.5 - In Exercises 1-8 determine whether the given graph...Ch. 10.5 - In Exercises 1-8 determine whether the given graph...Ch. 10.5 - Suppose that in addition to the seven bridges of...Ch. 10.5 - Prob. 10ECh. 10.5 - When can the centerlines of the streets in a city...Ch. 10.5 - Devise a procedure, similar to Algorithm 1, for...Ch. 10.5 - In Exercises 13-15 determine whether the picture...Ch. 10.5 - In Exercises 13-15 determine whether the picture...Ch. 10.5 - In Exercises 13-15 determine whether the picture...Ch. 10.5 - Show that a directed multigraph having no isolated...Ch. 10.5 - Show that a directed multigraph having no isolated...Ch. 10.5 - In Exercises 18-23 determine whether the directed...Ch. 10.5 - In Exercises 18-23 determine whether the directed...Ch. 10.5 - In Exercises 18-23 determine whether the directed...Ch. 10.5 - In Exercises 18-23 determine whether the directed...Ch. 10.5 - In Exercises 18-23 determine whether the directed...Ch. 10.5 - In Exercises 18-23 determine whether the directed...Ch. 10.5 - Devise an algorithm for constructing Euler...Ch. 10.5 - Devise an algorithm for constructing Euler paths...Ch. 10.5 - For which values of n do thesegraphs have an...Ch. 10.5 - For whichvalues ofndo the graphs in Exercise 26...Ch. 10.5 - For which values ofmandn.does the complete...Ch. 10.5 - Find the least number of times it is necessary to...Ch. 10.5 - In Exercises 30-36 determine whether the given...Ch. 10.5 - In Exercises 30-36 determine whether the given...Ch. 10.5 - In Exercises 30-36 determine whether the given...Ch. 10.5 - In Exercises 30-36 determine whether the given...Ch. 10.5 - In Exercises 30-36 determine whether the given...Ch. 10.5 - Prob. 35ECh. 10.5 - In Exercises 30-36 determine whether the given...Ch. 10.5 - Does the graph in Exercise 30 have a Hamilton...Ch. 10.5 - Does the graph in Exercise 31 have a Hamilton...Ch. 10.5 - Does the graph in Exercise 32 have a Hamilton...Ch. 10.5 - Does the graph in Exercise 33 have a Hamilton...Ch. 10.5 - Does the graph in Exercise 34 have a Hamilton...Ch. 10.5 - Does the graph in Exercise 35 have a Hamilton...Ch. 10.5 - Does the graph inExercise 36 have a Hamilton path?...Ch. 10.5 - For which values ofn.do the graphs in Exercise 26...Ch. 10.5 - For which values of m andndoes the complete...Ch. 10.5 - Show that thePetersen graph,shown here, does not...Ch. 10.5 - For each of these graphs, determine (i) whether...Ch. 10.5 - Can you find a simple graph with n vertices...Ch. 10.5 - Show that there is a Gray code of order whenever n...Ch. 10.5 - Fleury’s algorithm, published in 1883, constricts...Ch. 10.5 - Express Fleury's algorithm in pseudocode.Ch. 10.5 - Prob. 52ECh. 10.5 - Give a variant of Fleury's algorithm to produce...Ch. 10.5 - A diagnostic message can be sent out over a...Ch. 10.5 - Show that a bipartite graph with an odd number of...Ch. 10.5 - A knightis a chess piece that can move either two...Ch. 10.5 - A knightis a chess piece that can move either two...Ch. 10.5 - a) Show that finding a knights tour on...Ch. 10.5 - Show that there is a knight's tour on...Ch. 10.5 - Show that there is no knight's tour on...Ch. 10.5 - Show that there is no knight's tour on...Ch. 10.5 - Show that the graph representing the 1egal moves...Ch. 10.5 - Show that there is no reentrant knight's tour on...Ch. 10.5 - Show that there is a knight's tour on...Ch. 10.5 - The parts of this exercise outline a proof of...Ch. 10.5 - Show that if u and v are nondjacent vertices in a...Ch. 10.5 - Show that this graph doesnothave a Hamilton...Ch. 10.5 - Prob. 68ECh. 10.6 - For each of these problems about a subway system,...Ch. 10.6 - In Exercises 2-4 find the length of a shortest...Ch. 10.6 - In Exercises 2-4 find the length of a shortest...Ch. 10.6 - In Exercises 2-4 find the length of a shortest...Ch. 10.6 - Find a shortest path betweenaandzin each of the...Ch. 10.6 - Prob. 6ECh. 10.6 - Find shortest paths in the weighted graph in...Ch. 10.6 - Find a shortest path (in mileage) between each of...Ch. 10.6 - Find a combination of flights with the least total...Ch. 10.6 - Find a least expensive combination of flights...Ch. 10.6 - Find a shortest route (in distance) between...Ch. 10.6 - Find a routs with the shortest response time...Ch. 10.6 - Find a least expensive route, in monthly lease...Ch. 10.6 - Explain how to find a path mm the least number of...Ch. 10.6 - Exend Dijkstea's algorithm for finding the length...Ch. 10.6 - Extend Dijkstra's algorithm for finding the length...Ch. 10.6 - The weighted graphs in the figures here show some...Ch. 10.6 - Is a shortest path between two vertices in a...Ch. 10.6 - What are some applications where it is necessary...Ch. 10.6 - What is the length of a longest simple path in the...Ch. 10.6 - Floyd 's algorithm,displayed as Algorithm 2, can...Ch. 10.6 - Prove that Floyd's algorithm determines the...Ch. 10.6 - Give a big-0 estimate of the number of operations...Ch. 10.6 - Show that Dijkstra's algorithm may not work if...Ch. 10.6 - Solve the traveling salesperson problem for this...Ch. 10.6 - Solve the traveling salesperson problem far this...Ch. 10.6 - Find a route with the least total airfare that...Ch. 10.6 - Find a route with the least total airfare that...Ch. 10.6 - Construct a weighted undirected graph such that...Ch. 10.6 - Show that the problem of finding a circuit of...Ch. 10.6 - The longest path problemin a weighted directed...Ch. 10.7 - Can five houses be connected to two utilities...Ch. 10.7 - In Exercises 2-4 draw the given planar graph...Ch. 10.7 - In Exercises 2-4 draw the given planar graph...Ch. 10.7 - In Exercises 2-4 draw the given planar graph...Ch. 10.7 - In Exercises 5-9 determine whether the given graph...Ch. 10.7 - In Exercises 5-9 determine whether the given graph...Ch. 10.7 - In Exercises 5-9 determine whether the given graph...Ch. 10.7 - In Exercises 5-9 determine whether the given graph...Ch. 10.7 - In Exercises 5-9 determine whether the given graph...Ch. 10.7 - Complete the argument inExample 3.Ch. 10.7 - Show thatK5is nonplanar using an argument similar...Ch. 10.7 - Prob. 12ECh. 10.7 - Prob. 13ECh. 10.7 - Prob. 14ECh. 10.7 - ProveCorollary 3.Ch. 10.7 - Prob. 16ECh. 10.7 - Prob. 17ECh. 10.7 - Suppose that a planar graph haskconnected...Ch. 10.7 - Which of these nonplanar graphs have the property...Ch. 10.7 - Prob. 20ECh. 10.7 - In Exercises 20-22 determine whether the given...Ch. 10.7 - Prob. 22ECh. 10.7 - Prob. 23ECh. 10.7 - Prob. 24ECh. 10.7 - Prob. 25ECh. 10.7 - Prob. 26ECh. 10.7 - Prob. 27ECh. 10.7 - Prob. 28ECh. 10.7 - Prob. 29ECh. 10.7 - Show thatK3,3has 2 as its thickness.Ch. 10.7 - Find the thickness of the graphs in Exercise 27.Ch. 10.7 - Show that ifGis a connected simple graph...Ch. 10.7 - Prob. 33ECh. 10.7 - Prob. 34ECh. 10.7 - Prob. 35ECh. 10.7 - Prob. 36ECh. 10.7 - Draw K3,3on the surface of a torus so that no...Ch. 10.8 - Prob. 1ECh. 10.8 - Prob. 2ECh. 10.8 - Prob. 3ECh. 10.8 - Prob. 4ECh. 10.8 - Prob. 5ECh. 10.8 - Prob. 6ECh. 10.8 - Prob. 7ECh. 10.8 - Prob. 8ECh. 10.8 - Prob. 9ECh. 10.8 - Prob. 10ECh. 10.8 - Prob. 11ECh. 10.8 - Prob. 12ECh. 10.8 - Prob. 13ECh. 10.8 - What is the least number of colors needed to color...Ch. 10.8 - Prob. 15ECh. 10.8 - Show that a simple graph that has a circuit with...Ch. 10.8 - Schedule the final exams for Math 115, Math 116,...Ch. 10.8 - How many different channels are needed for six...Ch. 10.8 - The mathematics department has six committees,...Ch. 10.8 - Prob. 20ECh. 10.8 - Find the edge chromatic number of each of the...Ch. 10.8 - Prob. 22ECh. 10.8 - Find the edge chromatic numbers of a)Cn,wheren3....Ch. 10.8 - Prob. 24ECh. 10.8 - Show that ifGis a graph withnvertices, there no...Ch. 10.8 - Find the edge chromatic number ofKnwhen n is a...Ch. 10.8 - Prob. 27ECh. 10.8 - Prob. 28ECh. 10.8 - Construct a coloring of the graph shown using this...Ch. 10.8 - Use pseudocode to describe this coloring...Ch. 10.8 - Show that the coloring produced by this algorithm...Ch. 10.8 - Show thatCnis chromatically 3-critical whenevernis...Ch. 10.8 - Show thatWnis chromatically 4-critical whenever n...Ch. 10.8 - Prob. 34ECh. 10.8 - Prob. 35ECh. 10.8 - Find these values: a)X2(K3) b)X2(K4) c) X2(W4)...Ch. 10.8 - Prob. 37ECh. 10.8 - Prob. 38ECh. 10.8 - Frequencies for mobile radio (or cellular)...Ch. 10.8 - Show that every planar graph G can be colored...Ch. 10.8 - Prob. 41ECh. 10.8 - Show that g(3) = 1 and g(4) = 1 by showing that...Ch. 10.8 - Show that g(5) = 1. That is, show that all...Ch. 10.8 - Show that g(6) = 2by first using Exercises 42 and...Ch. 10.8 - Prob. 45ECh. 10.8 - Solve the art gallery problem by proving theart...Ch. 10 - a) Define a simple graph, a multigraph, a...Ch. 10 - Prob. 2RQCh. 10 - What is the relationship between the sum of the...Ch. 10 - Why must there be an even number of vertices of...Ch. 10 - Prob. 5RQCh. 10 - Describe the following families of graphs....Ch. 10 - Prob. 7RQCh. 10 - Prob. 8RQCh. 10 - a) Describe three different methods that can be...Ch. 10 - a) What does it mean for two simple graphs to be...Ch. 10 - a) What does it mean for a graph to be connected?...Ch. 10 - Prob. 12RQCh. 10 - a) Define an Euler circuit and an Euler path in an...Ch. 10 - Prob. 14RQCh. 10 - Give examples of at least two problems that can be...Ch. 10 - a) Describe Dijkstra's algorithm for finding the...Ch. 10 - a) What does it mean for a graph to be planar? b)...Ch. 10 - a) What is Eider's formula for connected planar...Ch. 10 - Prob. 19RQCh. 10 - a) Define the chromatic number of a graph. b) What...Ch. 10 - Prob. 21RQCh. 10 - Prob. 22RQCh. 10 - Prob. 1SECh. 10 - How many nonisomorphic subgraphs doesK3have?Ch. 10 - Prob. 3SECh. 10 - Prob. 4SECh. 10 - Prob. 5SECh. 10 - Prob. 6SECh. 10 - Prob. 7SECh. 10 - Prob. 8SECh. 10 - LetG= (V, E)be an undirected graph and let...Ch. 10 - Prob. 10SECh. 10 - Prob. 11SECh. 10 - Prob. 12SECh. 10 - Prob. 13SECh. 10 - Prob. 14SECh. 10 - We say that three verticesu, v, andwof a simple...Ch. 10 - Find the clustering coefficient of each of the...Ch. 10 - Prob. 17SECh. 10 - For each of the graphs in Exercise 17, explain...Ch. 10 - Prob. 19SECh. 10 - A cliquein a simple undirected graph is a complete...Ch. 10 - Prob. 21SECh. 10 - Prob. 22SECh. 10 - Prob. 23SECh. 10 - Prob. 24SECh. 10 - Prob. 25SECh. 10 - Prob. 26SECh. 10 - A simple graph can be used to determine the...Ch. 10 - A simple graph can be used to determine the...Ch. 10 - A simple graph can be used to determine the...Ch. 10 - A simple graph can be used to determine the...Ch. 10 - Prob. 31SECh. 10 - A simple graph can be used to determine the...Ch. 10 - Prob. 33SECh. 10 - Prob. 34SECh. 10 - Prob. 35SECh. 10 - Prob. 36SECh. 10 - An orientationof an undirected simple graph is an...Ch. 10 - Prob. 38SECh. 10 - Prob. 39SECh. 10 - A tournament is a simple directed graph such that...Ch. 10 - Prob. 41SECh. 10 - A tournamentis a simple directed graph such that...Ch. 10 - Prob. 43SECh. 10 - Prob. 44SECh. 10 - Prob. 45SECh. 10 - Prob. 46SECh. 10 - A connected graphG = (V, E)withnvertices and m...Ch. 10 - A connected graphG = (V, E)withnvertices and m...Ch. 10 - Prob. 49SECh. 10 - Prob. 50SECh. 10 - Prob. 51SECh. 10 - Thedistancebetween two distinct verticesv1and v2of...Ch. 10 - a) Show that if the diameter of the simple graph G...Ch. 10 - Prob. 54SECh. 10 - Prob. 55SECh. 10 - Devise an algorithm for finding the second...Ch. 10 - Prob. 57SECh. 10 - Prob. 58SECh. 10 - Show that ifGis a simple graph with at least 11...Ch. 10 - Prob. 60SECh. 10 - Prob. 61SECh. 10 - Show that the chromatic number of a graph is less...Ch. 10 - Suppose that to generate a random simple graph...Ch. 10 - For each of these properties, determine whether it...Ch. 10 - Prob. 65SECh. 10 - Prob. 66SECh. 10 - Prob. 1CPCh. 10 - Prob. 2CPCh. 10 - Prob. 3CPCh. 10 - Prob. 4CPCh. 10 - Prob. 5CPCh. 10 - Prob. 6CPCh. 10 - Prob. 7CPCh. 10 - Prob. 8CPCh. 10 - Given, a positive integer n, generate a simple...Ch. 10 - Prob. 10CPCh. 10 - Prob. 11CPCh. 10 - Prob. 12CPCh. 10 - Given the vertex pairs associated to the edges of...Ch. 10 - Given the ordered pairs of vertices associated to...Ch. 10 - Given the list of edges of a simple graph, produce...Ch. 10 - Given the list of edges of a simple graph, produce...Ch. 10 - Given the list of edges and weights of these edges...Ch. 10 - Given the list of edges of an undirected graph,...Ch. 10 - Prob. 19CPCh. 10 - Given the distances between pairs of television...Ch. 10 - Prob. 1CAECh. 10 - Prob. 2CAECh. 10 - Prob. 3CAECh. 10 - Prob. 4CAECh. 10 - Prob. 5CAECh. 10 - Prob. 6CAECh. 10 - Prob. 7CAECh. 10 - Prob. 8CAECh. 10 - Generate at random simple graphs with 10 vertices....Ch. 10 - Generate at random simple graphs with 10 vertices....Ch. 10 - Find the chromatic number of each of the graphs...Ch. 10 - Find the shortest path a traveling salesperson can...Ch. 10 - Prob. 13CAECh. 10 - Prob. 14CAECh. 10 - Describe the origins and development of graph...Ch. 10 - Prob. 2WPCh. 10 - Discuss the applications of graph theory to...Ch. 10 - Prob. 4WPCh. 10 - Explain what community structure is in a graph...Ch. 10 - Describe some of the algorithms used to detect...Ch. 10 - Prob. 7WPCh. 10 - Explain how graph theory can help uncover networks...Ch. 10 - Prob. 9WPCh. 10 - Prob. 10WPCh. 10 - Prob. 11WPCh. 10 - Prob. 12WPCh. 10 - Describe how Euler paths can be used to help...Ch. 10 - Prob. 14WPCh. 10 - Describe theChinese postman problemand explain how...Ch. 10 - Describe some of the different conditions that...Ch. 10 - Prob. 17WPCh. 10 - Prob. 18WPCh. 10 - In modeling, very large scale integration (VLSI)...Ch. 10 - Prob. 20WPCh. 10 - Prob. 21WPCh. 10 - Describe and compare several different algorithms...Ch. 10 - Explain how graph multicolorings can be used in a...Ch. 10 - Prob. 24WPCh. 10 - Explain how the theory of random graphs can be...
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- Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = = 2 kilograms, c = 8 kilograms per second, k 80 Newtons per meter, and F(t) = 20e¯* = Newtons. Solve the initial value problem. x(t) = = help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0 t→∞ ? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) ≈ x sp(t) = help (formulas) Book: Section 2.6 of Notes on Diffy Qsarrow_forwardConsider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and F(t) = 100 cos(8t) Newtons. Solve the initial value problem. x(t) = help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0 t→∞ ? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t)≈ x sp(t) = help (formulas) Book: Section 2.6 of Notes on Diffy Qsarrow_forwardConsider the initial value problem mx" cx' + kx F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 80 Newtons per meter, and F(t) = 20 sin(6t) kilograms, c = 8 kilograms per second, k = Newtons. Solve the initial value problem. x(t) = help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0 0047 ? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) ≈ x sp(t) = ☐ help (formulas) Book: Section 2.6 of Notes on Diffy Qsarrow_forward
- Consider the differential equation y' = - 4xy with initial condition y(0) = 1.9. Recall that Runge-Kutta method has the following formula for computing the next step, where h is the step size: f(xi, Yi) = fx i + (++) k1 = h k2 2 ¯‚ Yi + k₁ h h k3 = fxi 2 `, Yi + k₂· 2 k4 = f(xi+h, yikзh) i+1=i+h k12k22k3 + k4 Yi+1 Yi + h 6 Using Runge-Kutta step size h = 0.4: Estimate y(0.4) ≈ help (numbers) Estimate y(0.8) ≈ help (numbers) Book: Section 1.7 of Notes on Diffy Qsarrow_forwardDetermine which differential equation corresponds to each phase diagram. You should be able to state briefly how you know your choices are correct. х x 4 4 4 4 3 3 3 3 2 2 2 2 dx ? ✰ dt = 1. = x² - 3x 1 1 1 1 ? ◇ 2. dx dt = x(x − 2) - 0 0 0 0 ? ◇ 3. dx dt = x(2 − x)² -1 -1 -1 -1 Q -2 -2 -2 dx ? ◇ 4. ༤་ dt = = 3x - x² -3 -3 -3 -3 x³- 4x = x²|x − 2| ? ◇ 5. ம் dx dt བི་ dx ? ◇ 6. dt ཝེ་ dx ? 7. dt ཝེ་ dx ? ◇ 8. ཝེ་ dt -4 -4 -4 -4 A B 0 D = = 2x = x² * x * * x * K 4 4 4 4 = 4x - x³ 3 3 3 • 3 Book: Section 1.6 of Notes on Diffy Qs dit for this problem 2 2 2 2 1 1 1 1 0 0 0 8 -1 -1 -1 -1 N 心 -2 -2 -3 -3 -3 -4 -4 -4 -4 E FL G Harrow_forwardDear expert Chatgpt gives wrong answer Plz don't use chat gptarrow_forward
- An improved method that is similar to Euler's method is what is usually called the Improved Euler's method. It works like this: Consider an equation y' = f(x, y). From (xn, Yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step Xn+1 = xn +h, and then k1, the slope at (xn, Yn). The predicted new value of the solution . İs Zn+1 = Yn + h · k₁. Then we find the slope at the predicted new point k₁ = f(xn+1, Zn+1) and get the corrected point by averaging slopes h Yn+1 = = Yn + 1½ ½ (k1 + k₂). Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy dx = x - 0.5y, y(0.5) = 9. We let xo = 0.5 and yo 9 and pick a step size h = 0.25. Complete the following table: n xn Yn k1 Zn+1 k₂ 0 0.59-48 -3.25 ♡ <+ help (numbers) The exact solution can also be found for the linear equation. Write the answer as a function of x. y(x) = = help (formulas) Thus the actual value of the…arrow_forwardAlready got wrong Chatgpt answer If ur also Chatgpt user leave itarrow_forwardThe graph of the function f(x) is 1,0 (the horizontal axis is x.) Consider the differential equation x' = f(x). List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equalibria are stable, semi-stable (stable from one side, unstable from the other), or unstable. x = help (numbers) x = help (numbers) x = help (numbers) x = help (numbers) Book: Section 1.6 of Notes on Diffy Qsarrow_forward
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