2015
(May)
MATHEMATICS
(Major)
Course: 602
(Discrete Mathematics and Graph Theory)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions.
(A) DISCRETE MATHEMATICS
(Marks: 45)
1. Answer the following questions: 1x5=5
- The roots of the characteristic equation of a recurrence relation are
Find its general solution.
- It a lattice
, if
then
is it true?
- Give an example of an infinite lattice without
and
.
- Prove that in a complemented lattice
- A lattice L is given as
a
1 Is it a Boolean algebra?
2. Answer the following questions: 2x3=6
- Show that the following posets are not lattices.
- Show that the Lattice
, as given below is complemented.
1

0
- Find 5 (five sub lattices of
having 5 (five) elements.
3. Answer any two of the following questions: 3x2=6
- Solve the recurrence relation
- Show that every subset of a chain
is a lattice.
- For
being a Boolean algebra. Prove that
iff
.
4. Answer any two of the following questions: 5x2=10
- Define lattice homomorphism. Show that the given two lattices
and
are isomorphic. Here
is the power set of
with order relation
and
is the set of divisors of 30 with order relation
(divisors).
- Define minterm and maxterm with examples. Obtain the sum-of-product canonical form for the Boolean expression
- Represent the Boolean function
By its Karnaugh map and minimize.
Or
- Find a switching circuit for the logic expression
5. Answer any three of the following questions: 6x3=18
- Find the solution of the recurrence relation
- Define generating function of a sequence. If
is the
matrix
, evaluate
using recurrence relation and hence find
.
- Show that the minterms of the variables
have the properties
for
- Find the minimal form of the Boolean expression
- Implement
(B) GRAPH THEORY
(Marks: 35)
6. Answer the following questions: 1x3=3
- Give an example of pseudo graph.
- For the graph
given below, draw the sub graph
h
G= b g



f d
c
- What is Konigsberg’s Bridge problem?
7. Answer the following questions: 2x2=4
- Test isomorphism for the following graphs
and
- Show that the graph as shown below contains no Eulerian circuit:
8. Answer any two of the following questions: 5x2=10
- Prove that a simple graph with
vertices and
components can have at most
edges.
- State and prove handshaking theorem.
- Find the sum of the graphs
and
as given below:
9. Answer any three of the following questions: 6x3=18
- What is the difference between Eulerian and Hamiltonian graphs? For a single connected graph
with
vertices, if deg
+ deg
, for each vertices
and
not connected by an edge, then prove that
is Hamiltonian.
- Obtain simple graph, pseudo graph and multi graph from the following graph:
- Define adjacency matrix with example. Draw the graph represented by the following adjacency matrix:
- Prove that in a complete graph with
vertices, there are
edges disjoint Hamiltonian circuits, if
is an odd number
.
- What do you mean by linked representation of a graph? Draw adjacency structure for the graph given below: