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DEGREE OF A VERTEX

ISOLATED VERTEX: a vertex with zero degree

PENDANT: a vertex with degree one

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How many edges are there in a graph with ten vertices each of

degree six?

THEOREM 1

An undirected graph has an even number of vertices of odd degree

THEOREM 2

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DEGREE OF A VERTEX

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DEGREE OF A VERTEX OF DIRECTED GRAPH

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COMPLETE GRAPHS

Number of vertices = n

Number of edges = n(n-1)/2

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CYCLES

Number of vertices = n

Number of edges = n

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WHEELS

Wheelis obtained by adding an additional vertex to the cycle

and connect this new vertex to each of old vertices.

Number of vertices = n+1

Number of edges = 2n

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n-CUBES

Number of vertices =Number of edges =

n

2 12

nn

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COMPLETE BIPARTITE GRAPH

3,2K 5,3K

The graph that has its vertex set partitioned into two subsets of m

and n vertices, respectively. Thee is an edge between twovertices if and only if one vertex is in the first subset and other is

in the second subset.

Number of vertices = m+n

Number of edges = mn

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Are these graphs bipartite?

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COMPLEMENTARY GRAPH

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CONVERSE OF DIRECTED GRAPH

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INCIDENCE MATRIX

What is the sum of entries in a row of the incidence matrix?

What is the sum of entries in a column of the incidence matrix?

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INCIDENCE MATRIX

0

1

0

0

0

12

00000100100

00010010000

11001000000

00000001011

00100000000

1110987654321 e

e

d

c

b

a

eeeeeeeeeee

e1

e2

e3

e4

e5

e6

e7

e8

e9

e10

e11

e12

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ISOMORPHISM OF GRAPHS

The degrees of the vertices in isomorphic simple graphs must be same

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ISOMORPHISM OF GRAPHS

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ISOMORPHISM OF GRAPHS

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ISOMORPHISM OF GRAPHS

Subgraphs made up of the

vertices of degree 3 and the

edges connecting them must

isomorphic if the original

graphs are isomorphic.

ISOMORPHISM OF GRAPHS

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ISOMORPHISM OF GRAPHS

U1=V1

U2=V2U3=V8

U4=V5

U5=V6

U6=V7

U7=V4

U8=V10U9=V3

U10=V9

ISOMORPHISM OF GRAPHS

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ISOMORPHISM OF GRAPHS

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ISOMORPHISM OF GRAPHS

U1=V3

U2=V1

U3=V4U4=V2

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SELF-COMPLEMENTARY GRAPH

A simple graph is self complementary if G and its complementary are

isomorphic to each other.

a b

cd

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SELF-CONVERSE DIRECTED GRAPH

A directed graph is self-converse if it is isomorphic to its converse.

Are the following graphs self-converse?

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EULER CIRCUIT AND EULER PATH

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EULER CIRCUIT AND EULER PATH

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MOHAMMEDS SCIMITARS

EULER PATH

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EULER PATH

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THE KONIGSBERG BRIDGE PROBLEM

A

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HAMILTON PATH AND CIRCUIT

EULER AND HAMILTON CIRCUIT

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EULER AND HAMILTON CIRCUIT

If possible draw the Euler circuit and Euler path.

If possible draw the Hamilton circuit and Hamilton path.

INTERSECTION GRAPH

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INTERSECTION GRAPH

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PRECEDENCE GRAPH

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PRECEDENCE GRAPH

ROLL CT-2 ROLL CT-2 ROLL CT-2

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(10) (10) (10)

1 A 23 A 45 03

2 2.5 24 05 46 02

3 5.5 25 0.5 47 00

4 A 26 A 48 1.5

5 03 27 A 49 04

6 00 28 03 50 5.5

7 2.5 29 A 51 A

8 4.5 30 A 52 2.5

9 06 31 09 53 2.5

10 02 32 A 54 5.511 2.5 33 0.5 55 02

12 00 34 03 56 A

13 02 35 A 57 A

14 03 36 05 58 04

15 A 37 02 59 6.5

16 01 38 A 60 A

17 A 39 04 61 00

18 01 40 05 80 1.5

19 A 41 A 81 01

20 08 42 4.5 82 3.5

21 03 43 1.5 83 02

22 3.5 44 A 84 06

Cl t t 2 CSE 104

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1. a) S1 = {1,2,3}, R1 = {(1,1),(1,2),(2,3),(3,3)}. Find transitive closure of R1.

b) S2 = {1,2,3}, R2 = {(1,1),(1,3),(2,2),(3,2)}. Find transitive closure of R2.

c) S3 = {1,2,3}, R3 = {(1,2),(2,3),(2,2),(3,2)}. Find transitive closure of R3.

2. a) Is the function invertible? f1 = {(1,1),(2,3),(3,3),(4,2)}.

b) Is the function invertible? f 2= {(1,2),(2,3),(3,4),(4,4)}.c) Is the function invertible? f3 = {(1,1),(2,4),(3,3),(4,1)}.

3. a) Find cardinality of the set S = {x: x is a letter of the word benzene}

b) Find cardinality of the set S = {x: x is a letter of the word transitive}

c) Find cardinality of the set S = {x: x is a letter of the word impurity}

4. Give an example of a) Equivalence relation

b) Partial order relation

c) Antisymmetric relation [2+1+1+1]

Class test-2 CSE-104

Time: 30 minutes Marks:10