This is the topic of Discrete Structure.

1. DISCRETE STRUCTURE

2. GRAPH
TERMINOLOGIES
&
SPECIAL TYPE
GRAPHS

3. TYPES OF GRAPHS
Undirected Graphs
Directed Graphs
Special Simple Graphs
Special Simple Graphs

4. UNDIRECTED GRAPHS
The graph in which u and v(vertices)
are endpoints of an edge of graph G is
called an undirected graph G.
U V
LOOP

5. DEGREE
The number of edges for which vertex
is an endpoint.
The degree of a vertex v is denoted by
deg(v).

6. DEGREE
If deg(v) = 0, v is called isolated.
If deg(v) = 1, v is called pendant.

7. THE HANDSHAKING THEOREM
Let G = (V, E) be an undirected graph
with E edges.
Then
2|E| = vV deg(v)
Note that
This applies if even multiple edges and
loops are present.

8. EXAMPLE
How many edges are there in a graph
with 10 vertices each of degree 5?
o vV deg(v) = 6·10 = 60
o 2E= vV deg(v) =60
o E=30

9. EXAMPLE
How many edges are there in a graph
with 9 vertices each of degree 5?
o vV deg(v) =5 · 9 = 45
o 2E= vV deg(v) =45
o 2E=45
o E=22.5
o Which is not possible.

10. DIRCTED GRAPHS
When (u, v) is an edge of the graph G
with directed edges.
The vertex u is called the initial vertex
of (u, v), and v is called the terminal or
end vertex of (u, v).
The initial vertex and terminal vertex of
a loop are the same.

11. DEGREE
The in degree of a vertex v, denoted
deg−(v) is the number of edges which
terminate at v.
Similarly, the out degree of v, denoted
deg+(v), is the number of edges which
initiate at v.
vV deg－ (v) = vV deg＋ (v)= |E|

12. EXAMPLE
• Find the in-degree and out-degree of
each vertex in the graph G with
directed edges.
The Directed Graph G. 12

13. EXAMPLE
The in-degrees in G are
deg−(a) = 2 deg−(b) = 2
deg−(c) = 3 deg−(d) = 2
deg−(e) = 3 deg−(f ) = 0
The out-degrees in G are
deg+(a) = 4 deg+(b) = 1
deg+(c) = 2 deg+(d) = 2
deg+(e) = 3 deg+(f ) = 0

14. SOME SPECIAL SIMPLE GRAPHS
There are several classes of simple graphs.
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Complete graphs
Cycles
Wheels
n-Cubes
Bipartite Graphs
New Graphs from Old

15. COMPLETE GRAPHS
The complete graph on n vertices,
denoted by Kn, is the simple graph that
contains exactly one edge between
each pair of distinct vertices.
The Graphs Kn for 1≦ n ≦6.

16. COMPLETE GRAPHS
K5 & K6 is important because it is the
simplest non-planar graph.
It cannot be drawn in a plane with
nonintersecting edges.

17. CYCLE
The cycle Cn, n  3, consists of n
vertices v1, v2, . . ., vn and edges {v1, v2},
{v2, v3 } ,. . . , {vn-1, vn} , and {vn , v1}. The
cycles C3, C4, C5, and C6 are displayed
below.
The Cycles C3, C4, C5, and C6.

18. WHEEL
We obtain the wheel Wn when we add
an additional vertex to the cycle Cn, for
n  3, and connect this new vertex to
each of the n vertices in Cn, by new
edges.
The Wheels W3, W4, W5, and W6. 18

19. N-CUBES
Qn is the graph with 2n vertices
representing bit strings of length n.
An edge exists between two vertices
that differ by one bit position.
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20. EXAMPLE
A common way to connect processors
in parallel machines.
Intel Hypercube.

21. EXAMPLE
The n-cube Qn for n = 1, 2, and 3.
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22. BIPARTITE GRAPH
A simple graph G is called bipartite if
its vertex set V and be partitioned into
two disjoint sets V1 and V2 such that
every edge in the graph connects a
vertex in V1 and a vertex in V2 .
When this condition holds, we call the
pair (V1 , V2 ) a bipartition of the vertex
set V.

23. BIPARTITE GRAPH
 A Star network is a K(1,n) bipartite
graph.
 V1(n=ODD)
 V2(n=EVEN)

24. BIPARTITE GRAPH
V1={v1,v3,v5} ; V1={v2,v4,v6}
This is the graph of Hexagonal.

25. BIPARTITE GRAPH

26. NEW GRAPHS FROM OLD
A sub-graph of a graph G= (V, E) is a
graph H =(W, F), where W  V and F  E.
A sub-graph H of G is a proper sub-graph
of G if H  G .
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27. NEW GRAPHS FROM OLD
The graph G shown below is a sub-graph
of K5.
A Sub-graph of K5.
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28. NEW GRAPHS FROM OLD
The union of two simple graphs
G1= (V1, E1) & G2= (V2, E2)
is the simple graph with vertex set
V1  V2 and edge set E1  E2 .
The union of G1 and G2 is denoted by
G1  G2 .
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29. NEW GRAPHS FROM OLD
 The Simple Graphs G1 and G2
 Their Union G1∪G2.
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30. APPLICATION OF SPYCIAL TYPES OF GRAPHS
Suppose that there are m employees in
a group and j different jobs that need to
be done where m  j. Each employee is
trained to do one or more of these j
jobs. We can use a graph to model
employee capabilities. We represent
each employee by a vertex and each
job by a vertex. For each employee, we
include an edge from the vertex
representing that employee to the
vertices representing all jobs that the
employee has been trained to do.
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31. APPLICATION OF SPYCIAL TYPES OF GRAPHS
Note that the vertex set of this graph
can be partitioned into two disjoint sets,
the set of vertices representing
employees and the set of vertices
representing jobs, and each edge
connects a vertex representing an
employee to a vertex representing a job.
Consequently, this graph is bipartite.

32. APPLICATION OF SPYCIAL TYPES OF GRAPHS
Modeling the Jobs for Which Employees Have Been Trained.
To complete the project, we must
assign jobs to the employees so that
every job has an employee assigned to
it and no employee is assigned more
than one job.
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