Question #10 please
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question #10 please
![(a) Make a tree that shows all paths beginning at vertex
A. List the paths that terminate at C. Indicate which,
if any, are Hamiltonian.
(b) Is the graph Hamiltonian? Explain.
(c) Which roads should be paved so that one may drive
from A along paved roads to as many towns as pos-
sible at minimal cost? Justify your answer. What is
this minimal cost?
10. Solve the Traveling Salesman's Problem for each of the
following graphs by making trees that display all Hamil-
tonian cycles. (Start at A in each case.)
(а) А
(b)
25
30
10
10
20
\30
20
20
40
70
E
90 80
50
100
В
15
60
D
11. (a) Given that a tree has 100 vertices of degree 1 and
20 of degree 6 and that one-half of the remaining
vertices are of degree 4 and the rest of degree 2, de-
termine the number of vertices of degree 2.
(b) Prove that a tree as specified in (a), except that the
number of vertices of degree 1 must be less than 80,
cannot exist.
12. [BB] Suppose P1 and P2 are two paths from a vertex v to
another vertex w in a graph. Prove that the closed walk
obtained by following P1 from v to w and then P2 in
reverse from w to v contains a cycle.
13. [BB] (a) Draw the graphs of all nonisomorphic unla-
heled trees with five vertices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9e27b5d-0ab8-428e-954d-97d64fc14c61%2Fa5b94e96-20b8-4db9-844b-f9a5b4cee4be%2F8pn82pl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Make a tree that shows all paths beginning at vertex
A. List the paths that terminate at C. Indicate which,
if any, are Hamiltonian.
(b) Is the graph Hamiltonian? Explain.
(c) Which roads should be paved so that one may drive
from A along paved roads to as many towns as pos-
sible at minimal cost? Justify your answer. What is
this minimal cost?
10. Solve the Traveling Salesman's Problem for each of the
following graphs by making trees that display all Hamil-
tonian cycles. (Start at A in each case.)
(а) А
(b)
25
30
10
10
20
\30
20
20
40
70
E
90 80
50
100
В
15
60
D
11. (a) Given that a tree has 100 vertices of degree 1 and
20 of degree 6 and that one-half of the remaining
vertices are of degree 4 and the rest of degree 2, de-
termine the number of vertices of degree 2.
(b) Prove that a tree as specified in (a), except that the
number of vertices of degree 1 must be less than 80,
cannot exist.
12. [BB] Suppose P1 and P2 are two paths from a vertex v to
another vertex w in a graph. Prove that the closed walk
obtained by following P1 from v to w and then P2 in
reverse from w to v contains a cycle.
13. [BB] (a) Draw the graphs of all nonisomorphic unla-
heled trees with five vertices.
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