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.
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Due Sep 30, 11:59pm.
Consider the m by n grid graph:
n vertices in each of m rows, and m
vertices in each of n columns
arranged as a grid,
and edges between neighboring vertices on rows and columns
(excluding the wrap-around edges in the toric mesh).
There are m n vertices in total.
- What is the diameter of this graph?
- From the top left vertex to the bottom right vertex,
how many shortest paths are there? Please explain.
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Due Sep 30, 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
from D', we reverse all arcs of D. Using the idea of converse,
show that an acyclic digraph has a vertex with no outgoing arcs.
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Due Oct 7, 11:59pm.
DMM, §2.4, Problem 11 on page 59:
Suppose R is a matrix of 0's and 1's with 1's down the diagonal
(and perhaps elsewhere). Is R necessarily a reachability matrix
of some digraph? (Give proof or counterexample).
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Due Oct 7, 11:59pm.
Please draw the binary tree (with left-right children distinguished) corresponding
to the parenthesis expression
((())(())())(())(())().
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Due Oct 13, 11:59pm.
Suppose you have populations of 7000, 5000, 1000 in 3 states and n representatives
are to be allocated using Hamilton's method. There is an Alabama paradox
from n=6 to n=7. Please find a second pair (n,n+1) where a paradox is observed.