No. of Printed Pages : 5

MMTE-001

M.Sc. (MATHEMATICS WITH APPLICATIONS IN COMPUTER SCIENCE) (MACS) M.Sc. (MACS) Term-End Examination

00613

June, 2012 MMTE-001 : GRAPH THEORY Time : 2 hours

Maximum Marks : 50

Note : Question No. 1. is compulsory. Answer any four from the remaining six (2- 7). Calculators and similar devices are not allowed. 1.

2.

Are the following statements is true or false ? Give 10 reasons for your answers. (a)

If G is isomorphic to H, then the complement of G is isomorphic to the complement of H.

(b)

If the minimum degree 8 (G) > 2, then G contains a cycle.

(c)

If G has a spanning tree, then G is connected.

(d)

If X (G) = n, then G contains Kn as a subgraph.

(e)

Every Eulerian graph is 2 - connected.

(a) Draw a graph G with.

4

rad (G) < diam (G) < 2 rad (G). MMTE-001

1

P.T.O.

(b) Draw the dual of the following graph :

3

(c) For the following graph G find the following : (i)

(G) le (G)

(ii) A separating set S with ISI =

3.

K

(a) Draw a graph with six vertices and 9 edges which is (i)

4

(ii) Non planar

Planar

(b)

Draw a graph with n vertices, n —1 edges but having a cycle.

2

(c)

Use Havel-Hakimi theorem to check

4

whether the sequence {5, 5, 4, 4, 3, 3, 2, 2,} is graphic or not. If the sequence is graphical, construct a graph with the above degree sequence. MMTE-001

2

4.

(a) State with justification, whether the following graphs are isomorphic or not.

3

G:

H:

(b) Draw a 3 connected graph whose edge connectivity is 4 and minimum degree is 5. (c) Find the chromatic number of the Graph given below. If the chromatic number is k, give a k - colouring.

MMTE-001

3

3 4

5.

(a) Verify Brook's theorem for the following graph.

2

(b) State with justification whether the following graph is Hamiltonian or not.

3

0

0 0

0

0

0

Is it Eulerian, justify your answer. (c) Define a maximum matching and a perfect matching. Find a Maximum matching for the following graph G :

5

Does there exist a perfect matching for G ? Give justification. 6.

(a) Find the adjacency matrix and the incidence matrix of the following graph V2 e2

MMTE-001

4

4

P.T.O.

(b)

How many faces will a planer graph with degree sequence 3, 3, 3, 3, 3, 3, 6 will have ? Find the center of the following tree :

(c)

3

0

o

7.

3

(a) Prove that an edge e of a connected graph G is a cut edge if and only if e belongs to every spanning tree. (b) Find the minimum spanning tree for the following weighted graph using Prim's algorithm.

3

4

Q

(c) If G is disconnected, show that 6 - - is connected. Is the converse true ? Give justification.

MMTE-001

5

3

## Graph Theory.PDF

(c) Find the chromatic number of the Graph 4. given below. If the chromatic number is k, ... MMTE-001 3. Page 3 of 5. Main menu. Displaying Graph Theory.PDF.

#### Recommend Documents

Robust Graph Mode Seeking by Graph Shift
ample, in World Wide Web, dense subgraphs might be communities or link spam; in telephone call graph, dense subgraphs might be groups of friends or families. In these situations, the graphs are usually very sparse in global, but have many dense subgr

graph-contraction.pdf
... loading more pages. Retrying... graph-contraction.pdf. graph-contraction.pdf. Open. Extract. Open with. Sign In. Main menu. Displaying graph-contraction.pdf.

Graph Theory Notes - CiteSeerX
To prove the minimality of the set MFIS(X), we will show that for any set N ..... Prove that for a non-empty regular bipartite graph the number of vertices in both.

Knowledge Graph Identification
The web is a vast repository of knowledge, but automatically extracting that ... Early work on the problem of jointly identifying a best latent KB from a collec- ... limitations, and we build on and improve the model of Jiang et al. by including ....

Graph Algorithms
DOWNLOAD in @PDF Algorithms in C++ Part 5: Graph Algorithms: Graph Algorithms Pt.5 By #A#. Download EBOOK EPUB KINDLE. Book detail.

graph-search.pdf
There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. graph-search.

Polar Graph Paper.pdf
Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. Polar Graph Paper.pdf. Polar Graph Paper.pdf. Open. Extract.

Unbalanced Graph Cuts
tention recently, in the context of analyzing social networks and the World ..... [10]. We consider the converse of the densest k-subgraph problem, in which the.