## MA-351 Homework 2 Due at 4:59pm in my mailbox in SAS 3151, Tuesday, October 15, 2019

All solutions must be submitted in hardcopy either to me in class or placed in my mailbox. Note my office hours on my schedule.

1. 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.
1. What is the diameter of this graph?
2. From the top left vertex to the bottom right vertex, how many shortest paths are there? Please explain.

2. DMM, §2.3, Problem 15 on page 51. See Problem 13 for the definition of contrabasis.

3. DMM §3.6, Problem 13, page 168. Note that your coloring also needs to include the country "11" surrounding countries 1-10.

4. Suppose you have populations of 6000, 6000, 1000 in 3 states and n representatives are to be allocated using Hamilton's method. There is an Alabama paradox from n=5 to n=6 and another one from n=7 to n=8. Please find a third pair (n,n+1) where a paradox is observed.

5. Please draw the binary tree (with left-right children distinguished) corresponding to the parenthesis expression (())()(()())((())).