meine erste aussie klausur...

war...
(bitte meinung selbst bilden. angemerkt sei nur, dass das eine 3stuendige 3jahr-klausur war.)

Question 1:
a) Find the coordinates of the plane through the poins P(1,1,2,3), Q(3,4,7,5) and R(7,-3,4,11) in EE³ (= Extended Euclidean 3-space). Justify your answer.
b) Show that the points S(1,2,3,4), T(5,6,7,8), U(1,1,1,1) in EE² are collinear.

Question 2:
For n > 0 define the graph Q_n as follows: Vertices are vectors of length n with all components either 0 or 1. Two vertices are adjacent if they differ in exactly one component.
a) Draw Q_n for n = 1,2,3.
b) What are the order and the size of Q_n?
c) Show that Q_n is a regular graph; find its valency.
d) For which n is Q_n Eulerian?

Question 3:
a) Give the definition of a Hamiltonian graph.
b) Show that the graph Q_n from Problem 2 is Hamiltonian for n > 1.

Hint: Use mathematical induction on n and the fact that Q_n contains two disjoint copies of Q_(n-1).

Question 4:
a) Give the definition of a strongly regular graph.
b) Define \gamma as follows: Vertices are vectors of length 4, with all components either 0 or 1. Two vertices are adjacent if they differ in either 1 component or in all 4 components. Show that \gamma is a strongly regular graph with the following parameters:

v = 16, k = 5, \lambda = 0, \mu = 2