Question 1

How many different spanning trees can be found in the following network?

Accepted Answer

The answer of How many different spanning trees can be...

Question 2

Is it possible to have a tree in which every vertex has even degree? Explain.

Accepted Answer

No. A connected graph where ev

Question 3

The local town board has authorized the creation of a new sub-division consisting of five houses. The public works department is given the duty  of installing underground electric wires which will network together the houses with a nearby electric sub-station . The table below gives the coast in hundreds dollars of installing wires between each of the houses and sub-station . Find the coast og the cheapest network connecting the five houses with the sub-station.

Accepted Answer

A)  $3200 
B)  $3800 
C)  $4300 
D)  $5700 
E)  None of the above. 
A)  $3200 
B)  $3800 
C)  $4300 
D)  $5700 
E)  None of the above.

Question 4

What is the redundancy of the complete graph K25 ?

Accepted Answer

A)  275 
B)  276 
C)  300 
D)  25! - 1 
E)  None of the above. 
A)  275 
B)  276 
C)  300 
D)  25! - 1 
E)  None of the above.

Question 5

Consider the network shown below; draw two different spanning trees for the network.

Accepted Answer

Here are two possibl

Question 6

A connected graph contains 25 edges and every edge in the graph is a bridge. How many vertices does
the graph contain?

Accepted Answer

The answer of A connected graph contains 25 edges and...

Question 7

What number of edges will be bridges in a tree which has fifteen vertices?

Accepted Answer

The answer of What number of edges will be bridges...

Question 8

A city consists of five boroughs, and each borough has four neighborhoods and each neighborhood consists of ten houses. You have been hired to run a fiber-optic cable, networking all of the houses to each other. No matter the length, it costs $50 to connect any one house to any other house. Based on all of this information, what will it cost to network all of the houses together so that there are no redundancies? Hint: Think of each house as a vertex.

Accepted Answer

A)  $9950 
B)  $10500 
C)  $12500 
D)  $13500 
E)  None of the above. 
A)  $9950 
B)  $10500 
C)  $12500 
D)  $13500 
E)  None of the above.

Question 9

Consider the network shown below; what is the redundancy of the network?

Accepted Answer

The answer of Consider the network shown below; what is...

Question 10

Which of the following scenarios results in a graph which is not a tree?

Accepted Answer

A)  A graph with 5 vertices and 6 edges. 
B)  A graph in which every edge is a bridge. 
C)  A graph in which every pair of vertices has one unique path which connects them. 
D)  All of the above. 
E)  None of the above. 
A)  A graph with 5 vertices and 6 edges. 
B)  A graph in which every edge is a bridge. 
C)  A graph in which every pair of vertices has one unique path which connects them. 
D)  All of the above. 
E)  None of the above.

Question 11

Consider the network shown below; draw the minimum spanning tree for this network by applying Kruskal's algorithm.

Accepted Answer

Here is th

Question 12

The local town board has authorized the creation of a new sub-division of five houses. The public works department is given the duty of installing underground electric wires which will network together the houses with the nearby electric sub-station. The table below gives the cost in hundreds dollars of installing wires between each of the houses and the sub-station. Find the cost of the cheapest network connecting the five houses with the sub-station.

Accepted Answer

A)  $7100 
B)  $7200 
C)  $8800 
D)  $9600 
E)  None of the above. 
A)  $7100 
B)  $7200 
C)  $8800 
D)  $9600 
E)  None of the above.

Question 13

Consider the graph shown below; what is the weight of the minimum spanning tree of the graph?

Accepted Answer

A)   8.3 
B)   9.2 
C)   11.2 
D)   11.6 
E)  None of the above. 
A)   8.3 
B)   9.2 
C)   11.2 
D)   11.6 
E)  None of the above.

Question 14

A city consists of four boroughs, and each borough has six neighborhoods and each neighborhood consists of twenty houses. You have been hired to run a fiber-optic cable, networking all of the houses to each other. No matter the length, it costs $75 to connect any one house to any other house. Based on all of this information, what will it cost to network all of the houses together so that there are no
redundancies? Hint: Think of each house as a vertex.

Accepted Answer

The answer of A city consists of four boroughs, and...

Question 15

Consider Graph K shown below; which of the figures shown below are not spanning trees of Graph K?

Accepted Answer

A)  figure 1 
B)  figure 2 
C)  figure 3 
D)  Both figure 2 and figure 3 . 
E)  None of the above. 
A)  figure 1 
B)  figure 2 
C)  figure 3 
D)  Both figure 2 and figure 3 . 
E)  None of the above.

Question 16

Which of the following scenarios results in a graph which is a tree?

Accepted Answer

A)  A connected graph with 5 vertices and 4 edges. 
B)  A graph in which every edge is a bridge. 
C)  A graph in which every pair of vertices has one unique path which connects them. 
D)  All of the above. 
E)  None of the above. 
A)  A connected graph with 5 vertices and 4 edges. 
B)  A graph in which every edge is a bridge. 
C)  A graph in which every pair of vertices has one unique path which connects them. 
D)  All of the above. 
E)  None of the above.

Question 17

Consider the network shown in problem 4; how many edges are not bridges?

Accepted Answer

The answer of Consider the network shown in problem 4;...

Question 18

A city consists of three boroughs, and each borough has five neighborhoods and each neighborhood consists of ten houses. You have been hired to run a fiber-optic cable, networking all of the houses to each other. No matter the length, it costs $100 to connect any one house to any other house. Based on all of this information, what will it cost to network all of the houses in the city together so that there are
no redundancies? Hint: Think of each house as a vertex.

Accepted Answer

The answer of A city consists of three boroughs, and...

Question 19

Is every graph with 8 vertices and 7 edges a tree? Explain.

Accepted Answer

No. The gr

Question 20

Which of the graphs below are trees?

Accepted Answer

A)  Graph 1 
B)  Graph 2 
C)  Graph 3 
D)  Graph 2 and Graph 3 
E)  None of the above. 
A)  Graph 1 
B)  Graph 2 
C)  Graph 3 
D)  Graph 2 and Graph 3 
E)  None of the above.

How many different spanning trees can be found in the following network?

Is it possible to have a tree in which every vertex has even degree? Explain.

What is the redundancy of the complete graph K25 ?

Consider the network shown below; draw two different spanning trees for the network.

A connected graph contains 25 edges and every edge in the graph is a bridge. How many vertices does the graph contain?

What number of edges will be bridges in a tree which has fifteen vertices?

Consider the network shown below; what is the redundancy of the network?

Which of the following scenarios results in a graph which is not a tree?

Consider the network shown below; draw the minimum spanning tree for this network by applying Kruskal's algorithm.

Consider the graph shown below; what is the weight of the minimum spanning tree of the graph?

Consider Graph K shown below; which of the figures shown below are not spanning trees of Graph K?

Which of the following scenarios results in a graph which is a tree?

Consider the network shown in problem 4; how many edges are not bridges?

Is every graph with 8 vertices and 7 edges a tree? Explain.

Which of the graphs below are trees?

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The Mathematics of Touring: Traveling Salesman Problems

The Mathematics of Scheduling: Chasing the Critical Path

Population Growth Models: There Is Strength in Numbers

Financial Mathematics: Money Matters

The Mathematics of Symmetry: Beyond Reflection

Fractal Geometry: the Kinky Nature of Nature

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Censuses, Surveys, Polls, and Studies: the Joys of Collecting Data

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Exam 7: The Cost of Being Connected

How many different spanning trees can be found in the following network?

Is it possible to have a tree in which every vertex has even degree? Explain.

What is the redundancy of the complete graph K25 ?

Consider the network shown below; draw two different spanning trees for the network.

A connected graph contains 25 edges and every edge in the graph is a bridge. How many vertices does the graph contain?

What number of edges will be bridges in a tree which has fifteen vertices?

Consider the network shown below; what is the redundancy of the network?

Which of the following scenarios results in a graph which is not a tree?

Consider the network shown below; draw the minimum spanning tree for this network by applying Kruskal's algorithm.

Consider the graph shown below; what is the weight of the minimum spanning tree of the graph?

Consider Graph K shown below; which of the figures shown below are not spanning trees of Graph K?

Which of the following scenarios results in a graph which is a tree?

Consider the network shown in problem 4; how many edges are not bridges?

Is every graph with 8 vertices and 7 edges a tree? Explain.

Which of the graphs below are trees?

The Mathematics of Elections: the Paradoxes of Democracy

The Mathematics of Power: Weighted Voting

The Mathematics of Sharing: Fair-Division Games

The Mathematics of Apportionment: Making the Rounds

The Mathematics of Getting Around: Euler Paths and Circuits

The Mathematics of Touring: Traveling Salesman Problems

The Mathematics of Scheduling: Chasing the Critical Path

Population Growth Models: There Is Strength in Numbers

Financial Mathematics: Money Matters

The Mathematics of Symmetry: Beyond Reflection

Fractal Geometry: the Kinky Nature of Nature

Fibonacci Numbers and the Golden Ratio: Tales of Rabbits and Gnomons

Censuses, Surveys, Polls, and Studies: the Joys of Collecting Data

Graphs, Charts, and Numbers: the Data Show and Tell

Probabilities, Odds, and Expectations: Measuring Uncertainty and Risk

The Mathematics of Normality: the Call of the Bell

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