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mathematics
contemporary mathematics

Contemporary Mathematics 1st Edition OpenStax - Solutions

Draw a subgraph of Graph \(F\) from Figure 12.3 that has exactly four vertices and five edges. Graph G d hi g Multigraph H Figure 12.3 A Graph and a Multigraph
Which types of cycles are in Graph \(G\) in Figure 12.32? Use the vertices of the graph to give a name for one of each type that you find. Row 0 Row 1 Row 2 P Row 3 m Row 4 4 9 A Row 5 + 10 10 5 1 Row 6 1 6 15 20 15 6 1 Row 7 1 7 21 35 35 21 7 1 Row 8 8 28 56 70 56 28 8 Row 9 9 36 84 126 126 84 36
From Exercise 35 in Section 12.1 Exercises, Chloe is interested to know how many in her network of Roblox friends are also friends with each other, so she polls them. The following is a list of each of Chloe's friends and their common friends. Find a pentagon cycle in the graph of the Roblox
A sociologist is studying a community of 13 individuals in which every person has a social connection to every other person. How many closed triads are there in the community?
Name four differences between Graph \(C_{1}\) and Graph \(C_{2}\) that show they cannot be isomorphic. Graph C Graph C Two Distinct Graphs
Let's compare the games in Figures \(A\) and \(B\). In the game Diamonds, shown in Figure \(A\), each player begins in the space marked START and proceeds in a counterclockwise direction. On each turn, the player rolls a six-sided die and moves forward the number of spaces shown on the die. If the
. Graphs \(E\) and \(T\) are isomorphic. Name two isomorphisms. d Graph E Graph T Graph E and Graph T S
Three students have been asked to determine if Graphs \(E\) and \(T\) are isomorphic and justify their answers.Javier believes that Graphs \(E\) and \(T\) are not isomorphic because Graph \(E\) contains a triangle cycle and Graph \(T\) does not. He supports his conclusion with the graphs
Find the complement of the Graph \(H\). E D B Graph H
Suppose that a Graph \(M\) is a complete graph and Graph \(N\) is the complement of \(M\). What are the degrees of the vertices of Graph N? How do you know?
Recall from the Graph Basics section that a graph in which there are multiple edges between the same pair of vertices is called a multigraph. The figure shows a map of the four bridges that link Staten Island to Brooklyn and New Jersey and a multigraph representing the map. In the multigraph, the
Consider each sequence of vertices from Graph \(K\) in Figure 12.106 . Determine if it is only a walk, both a walk and a path, both a walk and a trail, all three, or none of these.1. \(n \rightarrow q \rightarrow 0 \rightarrow p\)2. \(p \rightarrow n \rightarrow q \rightarrow 0 \rightarrow n
Suppose that you want to travel from Palm Beach (PBI) to any other city listed in Figure 12.113 and then return to Palm Beach.1. Why is it impossible to find an itinerary that is a circuit?2. How is this related to the degree of the vertex PBI?
The given figure shows the same graph colored in several ways. The largest clique of the graph has three vertices. Use this information to answer each question.1. Is the graph planar? What does your answer tell you about the chromatic number?2. What is the lowest possible chromatic number of this
The first figure shows a map of the Island of Oahu in the State of Hawaii divided into regions, and the second figure shows a graph of the map. What is the chromatic number? Find a graph coloring that reflects the chromatic number of the graph. Kahuku Turtle Bay Sunset Beach Banzai Pipeline Shark's
Determine whether each graph is connected or disconnected and identify the set of vertices that make up each of its components. a b a b a b a ba b d d C d i Graph F Graph G Graph H Graph I Graph Graph K
"Six degrees of separation" is the idea that any two people on Earth are connected by at most six social connections. Assume that this is true. Consider a graph in which each vertex is a person on Earth, and each edge is a social connection. Would this graph be connected or disconnected? Explain
A pest control service has at least one customer on every block of every street or cul-de-sac in a neighborhood. Use the map of the neighborhood to answer each question.1. Draw a graph or multigraph of the neighborhood in which the vertices represent intersections, and the edges represent the
Use Graphs \(X\) and \(Y\) to answer each question.1. Which graph does not have an Euler circuit? Explain how you know.2. Find an Euler circuit in the other graph. b f f Graph X Graph Y
Create an eulerization using the fewest duplicate edges possible for Graph Z. a b C Graph Z Graph Z d
Use the figure to determine if each sequence of vertices represents an Euler trail or not. If not, explain why.1. \(e \rightarrow f \rightarrow b \rightarrow a \rightarrow d \rightarrow c \rightarrow g \rightarrow h \rightarrow e \rightarrow d\)2. \(d \rightarrow a \rightarrow b \rightarrow f
How many bridges and local bridges are in a complete graph with three or more vertices? Explain your reasoning.
Use Graph \(L\) to fill in the blanks to complete the steps in Fleury's algorithm.1. The two vertices that can be used as the starting vertex are ____________ and \(s\).2. If \(s q\) is the first edge removed, the three options for the second edge to be removed are \(q r\), ____________, and
Find an Euler circuit or trail through the graph using Fleury's algorithm. N W Graph T
Consider the figure shown. Is the circuit \(a \rightarrow b \rightarrow f \rightarrow e \rightarrow h \rightarrow i \rightarrow g \rightarrow d \rightarrow c \rightarrow a\) an Euler circuit, a Hamilton cycle, or both? a b C e h Graph T
Evaluate \(n\) ! and \((n-1)\) ! for \(n=6\).
Find the number of ways to arrange the letters \(v, w, x, y\), and \(z\).
How many Hamilton cycles are in the graph? W Complete Graph R
Find the weight of the Hamilton cycle \(m \rightarrow 0 \rightarrow p \rightarrow n \rightarrow q \rightarrow m\) in the given figure. 5 5, m 3 4 n 8 5' P 5 Weighted Complete Graph with Five Vertices
Use the graphs to determine if the given sequence of vertices represents a Hamilton path or not.1. Graph \(A, b \rightarrow c \rightarrow d \rightarrow e \rightarrow f\)2. Graph \(A, b \rightarrow e \rightarrow f \rightarrow c \rightarrow b \rightarrow d\)3. Graph \(K, n \rightarrow q \rightarrow 0
Use Figure 12.253 to find a Hamilton path between vertices \(C\) and \(E\).
Use Graph \(L\) to find a Hamilton path between each pair of vertices or indicate that there isn't one.1. \(p\) to \(r\)2. \(m\) to \(p\)3. o oto \(q\) Graph L S
Use Figure 12.262 to determine if the given sequence of vertices is a Hamilton path, an Euler trail, both, or neither.1. Graph \(A, a \rightarrow b \rightarrow e \rightarrow d \rightarrow c\)2. Graph \(F, g \rightarrow h \rightarrow i \rightarrow j \rightarrow h \rightarrow f \rightarrow g
Suppose that you lost the combination to a combination lock that consisted of three digits, and each was selected from the digits \(0,1,2,3,4,5,6,7,8\), or 9 . You desperately need to open the lock without breaking it. You decide to check all possible combinations methodically, 000 , then 001 ,
A complete weighted graph has vertices \(V, W, X, Y\), and \(Z\). List all the distinct Hamilton cycles. Use \(V\) as the starting vertex. Identify the pairs of cycles that have the same weight because they are reverses of each other.
.Suppose that the Air Force officer needed to leave from Travis Air Force base, visit each of Beal, Edwards, and Los Angeles Air Force bases exactly once and return to Travis. Use Figure 12.278 to find the shortest route.
Use the nearest neighbor method to find a Hamilton cycle of relatively low weight beginning and ending at vertex \(D\) in Figure 12.240 and find its total weight. Give your answer in hours and minutes.
There are some configurations that are commonly used when setting up computer networks. Several of them are shown in the given figure. Which of the configurations in the figure appear to have the characteristics of a tree graph? If a configuration does not appear to have the characteristics of a
Of the network configurations from Figure 12.297, which, if any, has the characteristics of a 1. Star tree?2. Caterpillar tree?3. Path graph?
Use Graphs \(I\) and \(/\) in Figure 12.301 to answer each question.1. Which vertices are in each of the components that remain when edge \(j\) / is removed from Graph \(ho\) ?2. Determine the number of edges and the number of vertices in Graph I. Explain how this confirms that Graph \(I\) is a
Use the given figure for the following exercises.1. Since \(s q\) is not an edge in Graph \(H\), Graph \(N_{1}\) cannot be a spanning tree of \(H\).a. Trueb. False 2. Graph \(\mathrm{N}_{2}\) is a spanning tree of Graph \(\mathrm{H}\).a. Trueb. False 3. Graph \(N_{3}\) is a spanning tree of Graph
Construct three distinct spanning trees for Graph \(J\). Graph J Graph
Name three edges that you could remove from Graph \(V\) in Figure 12.316 to form a third spanning tree, different from those in the solution to Example 12.50 Exercise 3.
Find a minimum spanning tree for the weighted graph. Give its total weight. 37 W 68 49 89 45 68 Weighted Graph 24
Which graphs, if any, are trees?Refer to the figure shown. q U e 9 e e g h
Which graphs, if any, are not trees because they are not connected?Refer to the figure shown. q U e 9 e e g h
Which graphs, if any, are not trees because they contain a cycle?Refer to the figure shown. q U e 9 e e g h
Tree graph Refer to the figure shown. Identify any graphs that fit the given description. d a a 9 h [m Graph T Graph T 9 a Graph T3 a Graph 74 Graph T5 Graph T6 b
Star graph Refer to the figure shown. Identify any graphs that fit the given description. d a a 9 h [m Graph T Graph T 9 a Graph T3 a Graph 74 Graph T5 Graph T6 b
Star like graph Refer to the figure shown. Identify any graphs that fit the given description. d a a 9 h [m Graph T Graph T 9 a Graph T3 a Graph 74 Graph T5 Graph T6 b
Line graph (or path graph)Refer to the figure shown. Identify any graphs that fit the given description. d a a 9 h [m Graph T Graph T 9 a Graph T3 a Graph 74 Graph T5 Graph T6 b
Lobster graph Refer to the figure shown. Identify any graphs that fit the given description. d a a 9 h [m Graph T Graph T 9 a Graph T3 a Graph 74 Graph T5 Graph T6 b
Caterpillar graph Refer to the figure shown. Identify any graphs that fit the given description. d a a 9 h [m Graph T Graph T 9 a Graph T3 a Graph 74 Graph T5 Graph T6 b
Forest graph Refer to the figure shown. Identify any graphs that fit the given description. d a a 9 h [m Graph T Graph T 9 a Graph T3 a Graph 74 Graph T5 Graph T6 b
Determine whether Graph \(H_{1}\) is a spanning tree of Graph \(H\). If not, explain how you know. a b C e -9
Determine whether \(\mathrm{Graph} \mathrm{H}_{2}\) is a spanning tree of Graph \(\mathrm{H}\). If not, explain how you know. e C 6 P
Determine whether Graph \(\mathrm{H}_{3}\) is a spanning tree of Graph \(H\). If not, explain how you know. a b d e 60
Determine whether Graph \(Q_{1}\) is a spanning tree of Graph \(Q\). If not, explain how you know. m n
Determine whether \(G\) raph \(Q_{2}\) is a spanning tree of Graph \(Q\). If not, explain how you know. J k [m
Determine whether Graph \(Q_{3}\) is a spanning tree of Graph \(Q\). If not, explain how you know. m n
How many more edges must be included with the dashed edges to create a spanning tree?A student has been asked to construct a spanning tree for Graph \(O\), as shown in the figure. The dashed lines show the first step that the student took, creating a path from vertex \(h\) to vertex \(d\). b a C e
List three unused (solid) edges from Graph \(O\) that cannot be used to complete the spanning tree.A student has been asked to construct a spanning tree for Graph \(O\), as shown in the figure. The dashed lines show the first step that the student took, creating a path from vertex \(h\) to vertex
Give an example of a set of edges that do not have \(e\) as an endpoint, which would complete the spanning tree.A student has been asked to construct a spanning tree for Graph \(O\), as shown in the figure. The dashed lines show the first step that the student took, creating a path from vertex
Give an example of a set of edges that do not have \(f\) as an endpoint, which would complete the spanning tree.A student has been asked to construct a spanning tree for Graph \(O\), as shown in the figure. The dashed lines show the first step that the student took, creating a path from vertex
How many more edges must be included with the dashed edges to create a spanning tree?A student has been asked to construct a spanning tree for Graph \(O\), as shown in the figure. The dashed lines show the first step that the student took, creating a path from vertex \(c\) to vertex \(h\). a d e h
List two unused edges from Graph \(O\) that cannot be used to complete the spanning tree.A student has been asked to construct a spanning tree for Graph \(O\), as shown in the figure. The dashed lines show the first step that the student took, creating a path from vertex \(c\) to vertex \(h\). a d
Give an example of a set of edges that do not have \(f\) as an endpoint, which would complete the spanning tree.A student has been asked to construct a spanning tree for Graph \(O\), as shown in the figure. The dashed lines show the first step that the student took, creating a path from vertex
Give an example of a set of edges that have neither \(c\) nor \(e\) as an endpoint, which would complete the spanning tree.A student has been asked to construct a spanning tree for Graph \(O\), as shown in the figure. The dashed lines show the first step that the student took, creating a path from
How many edges must be removed from Graph \(A\) to create a spanning tree?Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
How many edges must be removed from Graph \(B\) to create a spanning tree?Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
How many edges must be removed from Graph \(C\) to create a spanning tree?Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
Identify all the distinct cyclic subgraphs of Graph \(A\).Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
Identify all the cyclic subgraphs of Graph \(B\).Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
Identify all the cyclic subgraphs of Graph \(C\).Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
Draw four spanning trees of Graph \(A\) each of which includes edges \(v s, u v, w z\) and \(x y\).Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
Draw four spanning trees of Graph \(B\) which includes edge \(u t\), but not \(u r\).Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
Draw four spanning trees of Graph \(C\) that each have only one edge with an endpoint at vertex \(u\).Use Graphs \(A, B\), and \(C\). Graph A Graph B Graph C
\(S_{1}, S_{2}, S_{3}\) and \(S_{4}\) are all spanning trees.Use the figure shown. Draw a graph that fits the given description. a b a a b a b C 51 d 5 d C d 53 54
\(S_{1}\) and \(S_{2}\) are spanning trees, but \(S_{4}\) is not.Use the figure shown. Draw a graph that fits the given description. a b a a b a b C 51 d 5 d C d 53 54
\(S_{3}\) and \(S_{4}\) are spanning trees but \(S_{1}\) is not.Use the figure shown. Draw a graph that fits the given description. a b a a b a b C 51 d 5 d C d 53 54
\(S_{2}\) and \(S_{3}\) are spanning trees but \(S_{1}\) is not.Use the figure shown. Draw a graph that fits the given description. a b a a b a b C 51 d 5 d C d 53 54
\(S_{2}\) and \(S_{3}\) are spanning trees but \(S_{4}\) is not.Use the figure shown. Draw a graph that fits the given description. a b a a b a b C 51 d 5 d C d 53 54
\(S_{1}, S_{2}\), and \(S_{3}\) are spanning trees but \(S_{4}\) is not.Use the figure shown. Draw a graph that fits the given description. a b a a b a b C 51 d 5 d C d 53 54
\(S_{2}, S_{3}\), and \(S_{4}\) are spanning trees but \(S_{1}\) is not.Use the figure shown. Draw a graph that fits the given description. a b a a b a b C 51 d 5 d C d 53 54
\(S_{1}\) and \(S_{4}\) are spanning trees but \(S_{2}\) and \(S_{3}\) are not.Use the figure shown. Draw a graph that fits the given description. a b a a b a b C 51 d 5 d C d 53 54
Use the figure shown to find the weight of the given spanning tree. 13 3 S 1 2 4 6 t V 14 15 2 11 8 5 W 9 x y 10
Use the figure shown to find the weight of the given spanning tree. 13 3 S 1 2 4 6 t V 14 15 2 11 8 5 W 9 x y 10
Use the figure shown to find the weight of the given spanning tree. 13 3 S 1 2 4 6 t V 14 15 2 11 8 5 W 9 x y 10
Use Kruskal's algorithm to draw a minimum spanning tree for Graph \(Z\) in the provided figure. Find its weight.
Graph \(A\)Draw a minimum spanning tree for the given graph, and calculate its weight. 100 2.9 1.7 1.2 4 0 0 * d 30 3.1 3.5 4.3 40 Graph A Graph B Graph C Graph D
Graph \(C\)Draw a minimum spanning tree for the given graph, and calculate its weight. 100 2.9 1.7 1.2 4 0 0 * d 30 3.1 3.5 4.3 40 Graph A Graph B Graph C Graph D
Graph \(B\)Draw a minimum spanning tree for the given graph, and calculate its weight. 100 2.9 1.7 1.2 4 0 0 * d 30 3.1 3.5 4.3 40 Graph A Graph B Graph C Graph D
Graph \(D\)Draw a minimum spanning tree for the given graph, and calculate its weight. 100 2.9 1.7 1.2 4 0 0 * d 30 3.1 3.5 4.3 40 Graph A Graph B Graph C Graph D
City planners are tasked with building roadways to connect locations \(A, B, C\), and \(D\). The cost to build the roadways between any given pair of locations is given in the table.Draw a weighted graph to represent the given information. Then use the graph to find a minimum spanning tree and give
In a video game, the goal is to visit five different lands, \(V, W, X, Y\) and \(Z\), without losing all your lives. The paths between the lands are rated for danger, 1 being lowest and 10 being highest. Once a path has been traversed successfully, it is free from danger. The ratings are given in
Connected Use Graphs \(A, B, C, D, E, F, G, H\), and \(I\) to answer the following exercises. Identify any graph or graphs with the given characteristics or indicate that none do. b h g Graph A W Graph D a k P Graph B b h P k e C Graph E (m) d Graph G n Graph H Multigraph C t Graph F Multigraph I u
Disconnected Use Graphs \(A, B, C, D, E, F, G, H\), and \(I\) to answer the following exercises. Identify any graph or graphs with the given characteristics or indicate that none do. b h g Graph A W Graph D a k P Graph B b h P k e C Graph E (m) d Graph G n Graph H Multigraph C t Graph F Multigraph
Exactly two components Use Graphs \(A, B, C, D, E, F, G, H\), and \(I\) to answer the following exercises. Identify any graph or graphs with the given characteristics or indicate that none do. b h g Graph A W Graph D a k P Graph B b h P k e C Graph E (m) d Graph G n Graph H Multigraph C t Graph F
Exactly three components Use Graphs \(A, B, C, D, E, F, G, H\), and \(I\) to answer the following exercises. Identify any graph or graphs with the given characteristics or indicate that none do. b h g Graph A W Graph D a k P Graph B b h P k e C Graph E (m) d Graph G n Graph H Multigraph C t Graph F
Exactly four components Use Graphs \(A, B, C, D, E, F, G, H\), and \(I\) to answer the following exercises. Identify any graph or graphs with the given characteristics or indicate that none do. b h g Graph A W Graph D a k P Graph B b h P k e C Graph E (m) d Graph G n Graph H Multigraph C t Graph F

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