Define homogeneous recurrence relation. Write the first order and second order homogeneous recurrence relations with constant coefficients giving an example for each. Solve the following recurrence relation:  for  given that 

Define a recurrence relation. Describe the following problems with the help of examples which can be solved through Divide and Conquer technique and Show its recurrence relation.
(i) Binary Search
(ii) Merge Sort
Solve these recurrence relations with a substitution method

Verify, “If an undirected graph has exactly two vertices of odd degree there must be a path joining these two vertices.”

(a) Solve  for  by Substitution method.

(b) Solve the recurrence by using iterative approach :

a) Find a recurrence relation and initial conditions for 4,14,44,134, 404, …
b) Find the generating function of 2, 4, 8, 16, 32, ...

Define homogeneous recurrence relation. Write the first order and second order homogeneous recurrence relations with constant coefficients giving an example for each. Solve the following recurrence relation:  for  given that 

Prove the followings:
a) the sum of the degrees of the vertices of G is twice the number of edges
b) If W is a u-v walk joining two distinct vertices u and v, then there is a path joining u and v contained in the walk using the principles of mathematical induction
c) A connected graph G is Eulerian if and only if the degree of each of its vertices is even.
d) If G is a connected planar (p,q)-graph, then the number r of the regions of G is given by r = q - p +2

Is a Hamiltonian graph Eulerian ? Is a Eulerian graph Hamiltonian ? Show with the help of a suitable example.