Department of Systems Engineering and Engineering Management
The Chinese University of Hong Kong
LINEAR ALGEBRA & DISCRETE MATHEMATICS
SEG2410A/B
Exercise 5
Question 1.
Consider a tree with n vertices. It has exactly n-1 edges
[Lemma 2], so the sum of the degree of its vertices is 2n-2
[Theorem 3 of §6.1].
(a) A certain tree has two vertices of degree 4, one vertex of
degree 3 and one vertex of degree 2. If the other vertices have
degree 1, how many vertices are there in the graph? Hint: If the
tree has n vertices, n-4 of them will have to have degree 1.
(b) Draw a tree as described in part (a).
Question 2.
Count the number of spanning trees in the following graphs.
(a)
Question 3.
(b)
(c)
Consider a full m-ary tree with p parents and t leaves. Show that
t=(m-1)p+1 no matter what the height is.
Question 4.
(a) There are seven different types of rooted trees of height 2 in
which each node has at most two children. Draw one tree of
each type.
(b) To which of the types in part (a) do the regular binary trees of
height 2 belong?
(c) Which of the trees in part (a) are full binary trees?
Question 5.
Consider the graph shown in the followings:
V1
V2
V1
V2
V3
V3
V4
V4
V5
V5
V6
V6
V7
(a)
1)
2)
3)
4)
Is this a Hamiltonian graph?
Is this a complete graph?
Is this a bipartite graph?
Is this a complete bipartite graph?
(b)