Arrange the steps in the correct order to prove that a simple graph T is a tree if and only if it is connected, but the deletion of any of its edges produces a graph that is not connected. (First, prove that if a simple graph T is a tree, then it is connected and the deletion of any of its edges produces a graph that is not connected, and then prove the converse of it.) Rank the options below. Suppose that a simple connected graph 7 satisfies the condition that the removal of any edge will disconnect it. Suppose that T is a tree; then, by definition, it is connected. If T is not a tree, then it has a simple circuit, say, x1, x2, ..., xr, x1. 1 4 LQ 5 Now, T with (x, y) deleted has no path from x to y, since there was only one simple path from x to y in T, and the edge itself was it. Therefore, the graph with (x, y) deleted is not connected. 2 Let (x, y) be an edge of T such that x * y. This is a contradiction to the condition. Therefore, our assumption was wrong, and T is a tree. If we delete the edge (x, x1) from 7, then the graph will remain connected, since wherever the deleted edge was used in forming paths between vertices we can instead use the rest of the circuit x1, x2, ..., xr or its reverse. 6 ☑ × × 3 × 7 For which values of n do these graphs have an Euler circuit? 1 Wn 2 Kn 3 Сп 4 Qn Match each of the options above to the items below. n≥ 3 and n is odd. n≥ 3 No Euler circuit is possible for any value of n. n> 0 and n is even. 2 1 × × 4 × 3

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

Please help me with these questions. What are the correct Answers.

Thank you

Arrange the steps in the correct order to prove that a simple graph T is a tree if and only if it is connected, but the deletion of any of its edges produces a graph that is not connected. (First, prove that if a simple graph T is a tree, then it is
connected and the deletion of any of its edges produces a graph that is not connected, and then prove the converse of it.)
Rank the options below.
Suppose that a simple connected graph 7 satisfies the condition that the removal of any edge will disconnect it.
Suppose that T is a tree; then, by definition, it is connected.
If T is not a tree, then it has a simple circuit, say, x1, x2, ..., xr, x1.
1
4
LQ
5
Now, T with (x, y) deleted has no path from x to y, since there was only one simple path from x to y in T, and the edge itself was it. Therefore, the graph with (x, y) deleted is not connected.
2
Let (x, y) be an edge of T such that x * y.
This is a contradiction to the condition. Therefore, our assumption was wrong, and T is a tree.
If we delete the edge (x, x1) from 7, then the graph will remain connected, since wherever the deleted edge was used in forming paths between vertices we can instead use the rest of the circuit x1, x2, ..., xr
or its reverse.
6
☑
×
×
3
×
7
Transcribed Image Text:Arrange the steps in the correct order to prove that a simple graph T is a tree if and only if it is connected, but the deletion of any of its edges produces a graph that is not connected. (First, prove that if a simple graph T is a tree, then it is connected and the deletion of any of its edges produces a graph that is not connected, and then prove the converse of it.) Rank the options below. Suppose that a simple connected graph 7 satisfies the condition that the removal of any edge will disconnect it. Suppose that T is a tree; then, by definition, it is connected. If T is not a tree, then it has a simple circuit, say, x1, x2, ..., xr, x1. 1 4 LQ 5 Now, T with (x, y) deleted has no path from x to y, since there was only one simple path from x to y in T, and the edge itself was it. Therefore, the graph with (x, y) deleted is not connected. 2 Let (x, y) be an edge of T such that x * y. This is a contradiction to the condition. Therefore, our assumption was wrong, and T is a tree. If we delete the edge (x, x1) from 7, then the graph will remain connected, since wherever the deleted edge was used in forming paths between vertices we can instead use the rest of the circuit x1, x2, ..., xr or its reverse. 6 ☑ × × 3 × 7
For which values of n do these graphs have an Euler circuit?
1
Wn
2
Kn
3 Сп
4
Qn
Match each of the options above to the items below.
n≥ 3 and n is odd.
n≥ 3
No Euler circuit is possible for any value of n.
n> 0 and n is even.
2
1
×
×
4
×
3
Transcribed Image Text:For which values of n do these graphs have an Euler circuit? 1 Wn 2 Kn 3 Сп 4 Qn Match each of the options above to the items below. n≥ 3 and n is odd. n≥ 3 No Euler circuit is possible for any value of n. n> 0 and n is even. 2 1 × × 4 × 3
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,