MA-351 Homework 2
Due as indicated for each problem.
All solutions must be submitted on the Moodle web site
for the class at wolfware.ncsu.edu.
You may upload a photo of your handwritten solution or
a file of your typed solution.
Note my office hours on my schedule.
-
Due October 6, 11:59pm.
Please consider the n by n triangular grid graph:
Row
i, with
1
≦
i
≦
n,
has vertices
(
i,
j
)
in columns
j
with
i
≦
j
≦
n,
that is the grid does not extend to above the anti-diagonal.
There are edges
{
(i, j),
(i, j+1)
}
for all i with 1 ≦ i ≦ n–1 and j with i ≦ j ≦ n–1,
and
{
(i, j),
(i+1 ,j)
}
for all j with 2 ≦ j ≦ n and i with 1 ≦ i ≦ j–1,
- What is the diameter of this graph?
- From vertex (1,1) to vertex (n,n)
how many shortest paths are there? Please explain.
-
Due October 6, 11:59pm.
DMM, §2.3, Problem 12 on page 51:
The converse D' of a digraph D is defined as follows:
the vertex sets are the same, and (u,v) is an arc in D
if and only if (v,u) is an arc in D', that is, to
form D', we reverse all arcs of D. Using the idea of converse,
show that an acyclic digraph has a vertex with no outgoing arcs.
Hint: see the book page 47: link
-
Due Oct 13, 11:59pm.
DMM, §2.4, Problem 10 on page 59.
If G is a graph and R its reachability matrix:
(a) Show that R × R' = R.
Note that R' is the transposed of R and × is entry-wise
multiplication.
(b) What is the interpretation of the 1,1 entry in R2?
-
Due Oct 13, 11:59pm.
Please draw the binary tree (with left-right children distinguished) corresponding
to the parenthesis expression
((()))()(())((()))()()
-
Due Oct 18, 11:59pm.
Suppose you have populations of 7000, 7000, 3000 in 3 states and n representatives
are to be allocated using Hamilton's method. There is an Alabama paradox
from n=8 to n=9. Please find a second pair (n,n+1) where a paradox is observed.
