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| = vV 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 vV deg(v) = 6·10 = 60
o 2E= vV deg(v) =60
o E=30
9. EXAMPLE
How many edges are there in a graph
with 9 vertices each of degree 5?
o vV deg(v) =5 · 9 = 45
o 2E= vV 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.
vV deg- (v) = vV 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.
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)
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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