PGMT-8B (PT/10/VIIIB)
POST-GRADUATE COURSE
PGMT-8B (PT/10/VIIIB) 2
2.
a)
Let G be a simple graph with n vertices and
m components. Then prove that G can have
1
at most ( n − m )( n − m + 1 ) edges.
5
2
b)
State and solve the problem of Ramsey.
a)
Prove that a connected planar graph with n
vertices and e edges has ( e − n + 2 ) faces. 6
b)
Find the number of edges in a forest with n
vertices and m components.
4
a)
Define a rooted tree and level of its vertices.
What is an m-tree ? Prove that an m-tree
Term End Examination — December, 2013 / June, 2014
MATHEMATICS
Paper - 8B : Graph Theory
Time : 2 Hours
Full Marks : 50
( Weightage of Marks : 80% )
Special credit will be given for accuracy and relevance
in the answer. Marks will be deducted for incorrect
spelling, untidy work and illegible handwriting.
The marks for each question has been
indicated in the margin.
3.
4.
Answer Question No. 1 and any four from the rest.
1.
Answer any five questions :
2 × 5 = 10
a)
Define a simple graph and a regular graph
with example.
b)
What is handshaking lemma ? Explain.
c)
Differentiate between (i) Trail and Path,
(ii) Circuit and Cycle.
d)
Draw a graph having degree sequence
( 1, 2, 2, 4, 5 ).
e)
Show that every connected graph has at
least one spanning tree.
f)
Draw the complement of the graph K 2, 3 .
g)
If the degree of each vertex of a graph G is
at least 2, show that G contains a circuit.
PG-Sc.-6468-P
has at most m p vertices at level p.
[ P.T.O.
5.
5
b)
Prove that a graph G is a tree if and only if
there is a unique path between every pair of
vertices of G.
5
a)
Describe Kruskal's algorithm for finding a
minimal spanning tree of a connected
weighted graph with an example.
6
b)
Define connected graph. If any two distinct
vertices u and v of a simple graph with n
vertices are such that deg ( u ) + deg ( v ) ≥ n ,
then show that the graph is connected.
6.
5
a)
4
Let p denotes the number of vertices of a
tree T of degree 1 and q denotes the
number of vertices of degree ≥ 3. If
T contains at least two vertices, prove that
p ≥ q + 2 . Also show that p = q + 2 when
T does not contain any vertex of degree
greater than 3.
5
PG-Sc.-6468-P
3 PGMT-8B (PT/10/VIIIB)
b)
7.
a)
b)
Show that if a planar simple connected
graph contains 6 vertices and 12 edges,
then each of the faces is bounded by
3 edges.
5
Define
Graph-Homeomorphism
with
example. State the necessary and sufficient
condition for the planarity of a graph given
by Kuratowski.
3+2
Give the adjacency matrix of the following
graph G and put the matrix in a block
diagonal form like
5
PG-Sc.-6468-P