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mathematics
contemporary mathematics

Questions and Answers of Contemporary Mathematics

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
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
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
. 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 \(E\)
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
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
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
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
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
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
"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
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
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
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
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
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
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
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
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
.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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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

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