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mathematics
contemporary mathematics

Questions and Answers of Contemporary Mathematics

When a _________________ is removed from a graph, the shortest path between its vertices will be greater than two.Fill in the blank to make the statement true.
When using Fleury's algorithm to find an Euler trail, never remove a _________________ unless it is the only option.Fill in the blank to make the statement true.
A Hamilton cycle is a circuit that visits each _______________ exactly once.Fill in the blank to make the statement true.
A _______________ graph with \(n \geq 3\) vertices has ( \(n-1\) )! Hamilton cycles.Fill in the blank to make the statement true.
A Hamilton cycle _______________ a circuit.Fill in the blank with is or is not to make the statement true.
A Hamilton cycle that visits every edge _______________ an Euler circuit.Fill in the blank with is or is not to make the statement true.
A Hamilton cycle _______________ different from a Hamilton circuit.Fill in the blank with is or is not to make the statement true.
An Euler circuit that visits every vertex _______________ a Hamilton cycle.Fill in the blank with is or is not to make the statement true.
The total weight of a trail _______________ the sum of the weights of the edges visited by the trail.Fill in the blank with is or is not to make the statement true.
A weighted graph _______________ always a complete graph.Fill in the blank with is or is not to make the statement true.
The number of ways to arrange \(n\) objects _______________ \((n-1)!\)Fill in the blank with is or is not to make the statement true.
Every cycle _______________ a circuit.Fill in the blank with is or is not to make the statement true.
Unlike in a Hamilton cycle, the vertex where the Hamilton path begins is _______________ the vertex where the Hamilton path ends.Fill in the blank with the same as or different from to make the
If a sequence of vertices represents a Hamilton path, the number of vertices listed should be _______________ the number of vertices in the whole graph.Fill in the blank with the same as or different
To determine if a graph has a Hamilton path, use a method that is _______________ the method used to determine if a graph has an Euler trail.Fill in the blank with the same as or different from to
If a graph with a bridge has a Hamilton path, the starting vertex should be on the side of the bridge that is _______________ the side of the bridge with the ending vertex.Fill in the blank with the
A path between two vertices of a graph that visits each vertex of the graph exactly once is called an Euler path.a. Trueb. False
Any graph that has exactly two vertices of odd degree has a Hamilton path.a. Trueb. False
If a graph is composed of two cycles joined only at a single vertex \(p\), then no Hamilton path can be formed starting or ending at any vertex that is adjacent to \(p\).a. Trueb. False
If an edge \(a b\) is a bridge with at least three components on each side, then there is no Hamilton path between vertex \(a\) and any vertex on the other side of edge \(a b\).a. Trueb. False
The advantage of a greedy algorithm is that it is more efficient.a. Trueb. False
The disadvantage of a brute force algorithm is that it does not always give the ideal solution.a. Trueb. False
The nearest neighbor method is an example of a brute force algorithm.a. Trueb. False
The brute force method is an example of a greedy algorithm.a. Trueb. False
The brute force method is used to find a Hamilton cycle of least weight in a complete graph.a. Trueb. False
The nearest neighbor method is used to find the ideal solution to the traveling salesperson problem.a. Trueb. False
The traveling salesman problem involves finding the shortest route to travel between two points.a. Trueb. False
The traveling salesman problem can be represented as finding a Hamilton cycle of least weight on a weighted graph.a. Trueb. False
There is always more than one Hamilton cycle of least weight, a given Hamilton cycle and the reverse of that Hamilton cycle.a. Trueb. False
The greatest possible number of distinct weights for the Hamilton cycles of a complete graph with \(n\) vertices is ( \(n\) \(-1)!\)a. Trueb. False
The number of cycles in a spanning tree is one less than the number of vertices.a. Trueb. False
A spanning tree contains no triangles.a. Trueb. False
A spanning tree includes every vertex of the original graph.a. Trueb. False
There is a unique path between each pair of vertices in a spanning tree.a. Trueb. False
A spanning tree must be connected.a. Trueb. False
Kruskal's algorithm is a method for finding all the different spanning trees in a given graph.a. Trueb. False
Only graphs that are trees have spanning trees.a. Trueb. False
A minimum spanning tree of a given graph can be found using Kruskal's algorithm.a. Trueb. False
A minimum spanning tree of a given graph is the subgraph, which is a tree, includes every vertex of the original graph, and which has the least weight of all spanning trees.a. Trueb. False
If a graph contains any cut edges, they must be included in any spanning tree.a. Trueb. False
Name all the vertices and edges of graph \(F\) in Figure 12.5 . Graph F Figure 12.5 Graph F W
.Name all the pairs of vertices of graph Fin Figure 12.5 that are not adjacent. Graph F Figure 12.5 Graph F W
Determine the degree of each vertex of Graph J in Figure 12.7. If graph / represents direct flights between a set of airports, do any of the airports have direct flights to two or more of the other
A map of the Midwest is given in Figure 12.12. Create a graph of the region in which each vertex represents a state and each edge represents a shared border. North Dakota Minnesota South Dakota Iowa
Roblox is an online gaming platform. Chloe is interested to know how many people in her network of Roblox friends are also friends with each other so she polls them. Explain how a graph or multigraph
The graphs in Figure 12.17 represent neural networks, where the vertices are the nodes, and the edges represent functional connections between them. Which graph do you think would represent a network
Suppose that a graph has five edges.1. Find the sum of the degrees of the vertices.2. Draw two different graphs that demonstrate this conclusion.
Use the Sum of Degrees Theorem to determine the number of introductions required in a room with 1. 6 strangers.2. 10 strangers.3. \(n\) strangers.
In Figure 12.25, Graph \(G\) is given, along with four diagrams. Determine whether each diagram is or is not a subgraph of Graph \(G\) and explain why. b C Graph G e Diagram/ C Diagram K C b Diagram
Identify the types of cyclic subgraphs in Graph H in Figure 12.29 and name them. f b e d g C Graph H Figure 12.29 Graph H
The graphs in YOUR TURN 12.6 represent social communities. The vertices are individuals and the edges are social connections. Use the graphs to answer the questions.1. Which graph has the most
Use Pascal's Triangle in Figure 12.32 to find the number of triangles in a complete graph with 11 vertices. Row 0 Row 1 Row 2 1 2 Row 3 1 3 3 Row 4 4 6 4 7 Row 5 + + 10 10 5 Row 6 1 6 15 20, 15 6 1
Which of the three graphs in Figure 12.45 are isomorphic, if any? Justify your answer. X AA Graph B Graph B Graph B3 Figure 12.45 Three Similar Graphs
A teacher uses games to teach her students about colors and numbers as shown in Figure 12.48.In the Colors Game, shown in Figure \(A\), each player begins in the space marked START and proceeds in a
In Example 12.13, we showed that the Graphs \(B_{1}\) and \(B_{2}\) in Figure 12.45 are isomorphic. In Figure 12.54, labels have been assigned to the vertices of Graphs \(B_{1}\) and \(B_{2}\).
Determine whether Graphs \(G\) and \(S\) in Figure 12.56 are isomorphic. If not, explain how they are different. If so, name the isomorphism. a bc d e n mop Graph G Graph S Figure 12.56 Graph G and
A particular high school has end-of-course exams in (E3) English 3, (E4) English 4, (M) Advanced Math, (C) Calculus, (W) World History, (U) U.S. History, (B) Biology, and (P) Physics. No English 3
Use Figure 12.64 to answer each question.1. Find the complement of Graph \(K\).2. Identify an isomorphism between the complement of Graph \(K\) from part 1, and the complement of Graph H in YOUR TURN
Figure 12.74 shows the floor plan of a house. Use the floor plan to answer each question.1. Draw a graph to represent the floor plan in which each vertex represents a different room (or hallway) and
Consider each sequence of vertices from Graph \(A\) in Figure 12.79. 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. \(b \rightarrow c
Suppose that you need to travel by air from Miami (MIA) to Orlando (MCO) and you were restricted to flights represented on the graph. For the trip to Orlando, you decide to purchase tickets with a
In Example 12.17, we discussed a high school, which holds end-of-course exams in (E3) English 3, (E4) English 4, (M) Advanced Math, (C) Calculus, (W) World History, (U) U.S. History, (B) Biology, and
Find a coloring of the graph in Figure 12.99, which uses four colors or fewer. Use the resulting coloring as a guide to recolor the map in Figure 12.101. How many colors did you use? Does this
Use Figure 12.109 to answer each question.1. Find a path between vertex \(a\) and every other vertex on the graph, if possible.2. Identify all the components of Graph \(E\).3. Determine whether the
The U.S. Interstate Highway System extends throughout the 48 contiguous states. It also has routes in the states of Hawaii and Alaska, and the commonwealth of Puerto Rico. Consider a graph
A postal delivery person must deliver mail to every block on every street in a local subdivision. Figure 12.117 is a map of the subdivision. Use the map to answer each question.1. Draw a graph or
Use Figure 12.122 to answer each question.1. Verify the Graph Fis Eulerian.2. Find an Euler circuit that begins and ends at vertex \(c\). a au b Graph F Figure 12.122 Graph F
Use Graph \(A\) and multigraphs \(B, C, D\), and \(E\) given in Figure 12.129 to answer the questions.1. Which of the multigraphs are not eulerizations of Graph \(A\) ? Explain your answer.2. Which
Use Figure 12.132 to determine if each series of vertices represents a trail, an Euler trail, both, or neither. Explain your reasoning.1. \(a \rightarrow b \rightarrow e \rightarrow g \rightarrow f
Use the graph of a social network in Figure 12.140 to answer each question.1. Identify any bridges.2. If all bridges were removed, how many components would there be in the resulting graph?3.
Use Fleury's Algorithm to find an Euler trail for Graph \(J\) in Figure 12.143. b e Graph Figure 12.143 Graph/
Use Fleury's algorithm to find either an Euler circuit or Euler trail in Graph \(G\) in Figure 12.147. Figure 12.147 Graph G
Use Figure 12.161 to determine whether the given circuit is a Hamilton cycle, an Euler circuit, both, or neither.1. \(a \rightarrow b \rightarrow c \rightarrow e \rightarrow h \rightarrow g
Evaluate \(n\) ! and \((n-1)\) ! for \(n=4\).
Find the number of ways to arrange the letters \(a,b, c\), and \(d\).
How many Hamilton cycles are in the complete graph in Figure 12.163? Figure 12.163 Complete Graph L
Use Figure 12.164 and the given Hamilton cycles to answer the following questions.\(V \rightarrow L \rightarrow E \rightarrow B \rightarrow V\)\(V \rightarrow L \rightarrow B \rightarrow E
Which of the following sequences of vertices is a Hamilton path for Graph \(Q\) in Figure 12.168?1. \(a \rightarrow d \rightarrow b \rightarrow c \rightarrow e \rightarrow g \rightarrow f\)2. \(c
Use Figure 12.177 to find a Hamilton path between vertices \(C\) and \(D\). B Figure 12.177 Graph G
Find a Hamilton path from vertex \(s\) to vertex \(v\) for each graph in Figure 12.179 or indicate that there is none. W W Graph A Graph B V W Graph C Graph D Figure 12.179 Graphs A, B, C, and D
Use Figure 12.185 to determine if the given sequence of vertices is a Hamilton path, an Euler trail, both, or neither.1. Graph \(A, e \rightarrow b \rightarrow a \rightarrow e \rightarrow d
A cashier rings up a sale for \(\$ 4.63\) cents in U.S. currency. The customer pays with a \(\$ 5\) bill. The cashier would like to give the customer \(\$ 0.37\) in change using the fewest coins
Suppose you have a complete weighted graph with vertices \(N, M, O\), and \(P\).1. Use the formula \((n-1)\) ! to calculate the number of distinct Hamilton cycles in the graph.2. Use the formula
On the next assignment, the air force officer must leave from Travis Air Force base, visit Beal, Edwards, and Vandenberg Air Force bases each exactly once and return to Travis Air Force base. There
Suppose that the candidate for governor wants to hold rallies around the state but time before the election is very limited. They would like to leave their home in city \(A\), visit cities \(B, C, D,
Identify any trees in Figure 12.205. If a graph is not a tree, explain how you know. 20 b f d g Graph M h Graph N m Figure 12.205 Graphs M, N, and P S t r 9 Graph P
Each graph in Figure 12.208 is one of the special types of trees we have been discussing. Identify the type of tree. e 9 h m n C S V W r u a Graph U Graph V Figure 12.208 Graphs U and V
Use Graphs \(I\) and \(J\) in Figure 12.212 to answer each question.1. Which vertices are in each of the components that remain when edge be is removed from Graph \(I\) ?2. Determine the number of
Use Figure 12.215 to determine which of graphs \(M_{1}, M_{2}, M_{3}\), and \(M_{4}\), are spanning trees of \(Q\). a a a b e e e e e Graph Q Graph M Graph M Graph M3 Graph M4 Figure 12.215 Graphs Q,
Construct two distinct spanning trees for the graph in Figure 12.220. Graph L Figure 12.220 Graph L
Use the graph in Figure 12.225 to answer each question.1. Determine the number of edges that must be removed to reveal a spanning tree.2. Name all the undirected cycles in Graph \(V\).3. Find two
A computer network will be set up with six devices. The vertices in the graph in Figure 12.229 represent the devices, and the edges represent the cost of a connection. Find the network configuration
Name all the vertices and edges of Graph \(A\). S Graph A Graph A 9 t
Name all the pairs of vertices of graph \(A\) in Figure 12.6 that are not adjacent. Iceland Norway Sweden Finland Estonia and United Kingdom Germy Portugal France Andorra Moscow Copenhagen (CPH) X Rd
Name a vertex of Graph \(A\) in Figure 12.6 with degree 4. Iceland Norway Sweden Finland and United Kingdom Germy Portugal France Andorra Estonia Moscow Copenhagen (CPH) X Romania Ukraine Turkey
The figure shows a map of the Island of Oahu in the State of Hawaii divided into regions. Draw a graph in which each vertex represents one of the regions and each edge represents a shared land
In a particular poker tournament, five groups of five players will play at a table until one player wins, then the five winning players will play each other at a table in a final round. Explain how a
Sociologists use graphs to study connections between people and to identify characteristics that make communities more resilient, more likely to stay connected over time. In the graphs shown, the
Suppose that the sum of the degrees of a graph is six.1. Find the number of edges.2. Draw two graphs that demonstrate your conclusion.
Determine the number of introductions necessary in a room with 500 strangers using the Sum of Degrees Theorem.

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