# Handouts

A Maple 11 worksheet on Coloring Cubes:  http://www.nd.edu/~sommese/math08S610/ColorCubes.mws

A Maple 11 worksheet for counting four vertex graphs: http://www.nd.edu/~sommese/math08S610/FourVertexGraphs.mws

A Maple 11 worksheet for the Petersen graph: http://www.nd.edu/~sommese/math08S610/PetersenGraph.mws

# Homework

## Homework 9 (due Friday, April 25):

A) Draw the trees with Prufer encodings

1234567

1111111

222

111333

B) Do a cyclic permutation on the vertices of the graphs you have constructed, e.g.,

for the first tree above (on the 9 vertices 1, 2, 3, 4, 5, 6, 7, 8, 9), replace 1 by 2,

2 by 3, etc.  Now compute the Prufer encodings of these permuted graphs.

C) What are the degrees of the 36 vertices of the graph with Prufer encoding equal to 13 ones

followed by 12 twos followed by 6 ones followed by 3 nines.

## Homework 9 (due Wednesday, April 16):

Compute the chromatic polynomial for the graphs K_n with n from 1 to 5;  and

for the graphs K_{a,b} with

## Homework 8 (due Wednesday, April 2):

For the graphs K_n with n from 3 to 8; for K_3 + K_5

joined at one point; and for K_4+K_4 joined at one point,

a) write down the adjacency matrices;

b) compute the number of edges;

c) compute the number of walks of length 4;

d) compute the number of triangles;

e) compute the ranks of the adjacency matrices;

f) compute the eigenvalues of the matrices; and

g) compute the diameters of the graphs.

## Homework 7 (due Wednesday, March 26):

Given a poset P = (X, R) with |X| = n > 0; we have shown there is at least one

order-preserving map to a chain of length n.  For each of the posets P on page 188,

use the zeta functions of their associated lattices L(P) to compute the numbers of

these order preserving maps.

## Homework 7 (due Wednesday, March 19):

Problems 1 and 8 of 12.10.

## Homework 6 (due Wednesday, March 12):

Problems 8 of 7.5.

## Homework 5 (due Wednesday, February 27):

Problems 1 (only for faces and for edges), 2 of 15.8.

## Homework 4 (due Wednesday, February 13):

Problems 2, 3, 6, and 7 of 13.6.

A) Write the permutation  in cycle notation.  This gives a partition of 8.

B) How many different permutations give rise to the same partition?

## Homework 3 (due Friday, February 8):

Problems 15b, 16 of 4.8.

Do Problems 7 and 8 of 5.6.

## Homework 2 (due Friday, February 1):

Compute S(n,k) directly for k greater than n - 3 and also for k less than 3.

Do Problems 14, 17 of 4.8.

## Homework 1 (due Friday, January 25):

Do Problems 3, 8, 10, 11 of 2.8.

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