Soheila Molaei
:According to factorization on conditional independence
.1.1
:If we choose f(x,y)=P(x|z) and g(y,z)=P(z)P(y|z) so
:Now for other direction
For , let and
.and hence so this proves conditional independence
.1.2
:Prov e or di s prov e .1.3
.1

Math 412
HW6
3/08/2014
Name:
Due Friday, March 14, 2014
Students in section X13 (three credit hours) need to solve any four of the following five
problems. Students in section X14 (four credit hours) must solve all five problems.
1. # 3.1.23 in the book
2

Probabilistic Graphical Models: Homework 2
Due April 28th, (Ordibehesht 9th), beginning of class
April 17, 2013
Instructions: There are 3 questions on this assignment. Please submit your homework in
the same order they appeared in the homework.
1
Conditio

Math 412
HW8
4/06/2014
Name:
Due Friday, April 11, 2014
Students in section X13 (three credit hours) need to solve any four of the following five
problems. Students in section X14 (four credit hours) must solve all five problems.
1. Let G be a n-vertex 3-

Math 412
HW3
Due Friday, February 19, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. For k 2, prove that every k-regular bipartite graph has no cut

Math 412
HW4
2/14/2014
Name:
Due Friday, February 21, 2014
Students in section X13 (three credit hours) need to solve any four of the following five
problems. Students in section X14 (four credit hours) must solve all five problems.
1. Prove that each of

Math 412
HW7
Due Friday, April 1, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. Let G be a bipartite graph. Prove that if (G) > n(G)/4, then 0 (G)

Math 412
HW9
4/12/2014
Name:
Due Friday, April 18, 2014
Students in section X13 (three credit hours) need to solve any four of the following five
problems. Students in section X14 (four credit hours) must solve all five problems.
1. # 4.3.3 in the book.
2

Math 412
HW4
Due Friday, February 26, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. Let T, T 0 be spanning trees of a connected graph G. For any e

Math 412
HW9
Due Friday, April 29, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. Prove that if G is a color critical graph, then the graph G0 gene

Math 412
HW6
Due Wednesday, March 16, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. A standard card deck has 52 cards with 4 suits (hearts, spades

Math 412
HW11
5/01/2014
Name:
Due Wednesday, May 07, 2014
Students in section X13 (three credit hours) need to solve any three of the following four
problems. Students in section X14 (four credit hours) must solve all four problems.
1. Use Kuratowskis The

Math 412
HW5
Due Friday, March 4, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. Prove that a d-regular simple graph G has a decomposition into cop