5. (a) Consider the trees with 17 vertices which have exactly 3 leaves. (i) Prove that any such tree must have a unique vertex of degree 3. (ii) Find the number of isomorphism classes of such trees in which the unique vertex of degree 3 is adjacent to a leaf. (b) A tree has a vertex of degree 3 and a vertex of degree 4. What is the minimum possible number of vertices in such a tree? Justify your answer. (c) Calculate the number of spanning trees for the following graph and explain your calculation: 6 D 5 2 3

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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5.
(a) Consider the trees with 17 vertices which have exactly 3 leaves.
(i) Prove that any such tree must have a unique vertex of degree 3.
(ii)
Find the number of isomorphism classes of such trees in which the
unique vertex of degree 3 is adjacent to a leaf.
(b) A tree has a vertex of degree 3 and a vertex of degree 4. What is the
minimum possible number of vertices in such a tree? Justify your answer.
(c) Calculate the number of spanning trees for the following graph and explain
your calculation:
6,
2
5
3
4
Transcribed Image Text:5. (a) Consider the trees with 17 vertices which have exactly 3 leaves. (i) Prove that any such tree must have a unique vertex of degree 3. (ii) Find the number of isomorphism classes of such trees in which the unique vertex of degree 3 is adjacent to a leaf. (b) A tree has a vertex of degree 3 and a vertex of degree 4. What is the minimum possible number of vertices in such a tree? Justify your answer. (c) Calculate the number of spanning trees for the following graph and explain your calculation: 6, 2 5 3 4
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