Home Work 8 (Discrete)
1- Find a graph with degree sequence degn = {3, 3, 3, 3, 2, 2}. Represent this graph by
adjacency and incidence matrices.
A
B
E1
E3
C
E2
E6
E4
E7
E5
D
E8
E
F
Adjacency Matrices: We order the vertices as A, B, C, D, E, F. The adjacency matrix
representing this graph is
𝟎
𝟏
𝟎
𝟏
𝟏
[𝟎
𝟏
𝟎
𝟏
𝟏
𝟎
𝟎
𝟎
𝟏
𝟎
𝟎
𝟏
𝟏
𝟏
𝟏
𝟎
𝟎
𝟎
𝟎
𝟏
𝟎
𝟏
𝟎
𝟎
𝟏
𝟎
𝟎
𝟏
𝟎
𝟏
𝟎]
Incidence Matrices: We order the vertices as A, B, C, D, E, F. and the edges as E1, E2,
E3, E4, E5, E6, E7, E8. The Incidence matrix representing this graph is
𝟏
𝟏
𝟎
𝟎
𝟎
[𝟎
𝟎
𝟏
𝟏
𝟎
𝟎
𝟎
𝟏
𝟎
𝟎
𝟏
𝟎
𝟎
𝟎
𝟏
𝟎
𝟏
𝟎
𝟎
𝟏
𝟎
𝟎
𝟎
𝟏
𝟎
𝟎
𝟎
𝟏
𝟎
𝟏
𝟎
𝟎
𝟎
𝟏
𝟎
𝟎
𝟏
𝟎
𝟎
𝟎
𝟎
𝟏
𝟏]
2- Is it possible to construct a graph with degree sequence degn ={ 3,3,3,2,2,1,1}?
No,
deg(vi )=2m which is even
(Note that this applies even if multiple edges and loops are present)
3+3+3+2+2+1+1=15 odd (not even)
3- List all (not isomorphic) connected graphs with 4 vertices.
4- Let T be a tree with 5 vertices of degree 3, 5 vertices of degree 4, and 5 vertices of
degree 5. All other vertices are of degree 1. Find the number of the leaves (vertices of
degree1) in the tree T using hand-shaking theorem.
-
Let L be the number of leaves in the tree T.
The number of edges (E) in T is n-1 (Theorem2 CH11.1) where n is the number
of vertices:
E = n-1 = (5 + 5 + 5 + L) -1
-
𝑑𝑒𝑔(𝑣𝑖 ) = 2𝐸
From the hand-shaking theorem it follows 𝑑𝑒𝑔(𝑣𝑖 )=(5*3 + 5*4 + + 5*5 + L*1)
= 2*(5+5+5+L-1). After Solving this equation:
L=32
T contains 32 leaves.
5- Find in which order are the vertices of the ordered rooted tree visited using a
preorder, inorder and postorder traversal.
Preorder: a, b, e, j, k, n, o, p, f, c, d, g, l, m, h, i.
Inorder: j, e, n, k, o, p, b, f, a, c, l, g, m, d, h, i.
Postorder: j , n, o, p, k, e, f , b, c, l, m, g, h, i, d, a.
Note: See example 2, 3 and 4 in text book (11.3).