graphs

1.  A Graph is a non-linear data
structure consisting of nodes and
edges.
 The set of edges describes
relationships among the vertices.
 So, A Graph is a finite set of
vertices(or nodes) and set of Edges
which connect a pair of nodes.
Graphs

2.  A graph G is defined as follows:
G=(V,E)
V(G): a finite, nonempty set of
vertices
E(G): a set of edges (pairs of
vertices)
Graphs

3. Example : Graph G
Graphs
A B C
D E
Vertices V(G) = {A, B, C , D, E}
Edges E(G) = { (A,B), (A,D), (B,C), (B,D), (C,E),
(D,E)}

4. According to Path or Flow:
 Undirected: In a graph if edges
have no direction then it is called
Undirected Graph.
 Directed: In a graph if edges have
direction then it is called Directed
Graph.
Graphs Types

5. Graphs Types
A B C
D E Undirected Graph
Vertices V(G) = {A, B, C , D, E}
Edges E(G) = { (A,B), (A,D), (B,C), (B,D), (C,E),
(D,E)}

6. Graphs Types
Directed Graph
A
B C
D E
Vertices V(G) = {A, B, C , D, E}
Edges E(G) = { (A,B), (A,D), (B,C), (D,B), (D,E),
(E,C)}

7. Graph Terminology
Adjacent Nodes: Two nodes are
adjacent if they are connected
with an edge.
A is Adjacent to B
B is adjacent From A
Path: A sequence of vertices
that connect two nodes in a
graph.
A B

8. Graph Terminology
Complete Graph: A graph in
which every vertex is directly
connected to every other
vertex.
if Number of vertices (v) are n then
Number of edges (e) are
In Directed Graph e=n*(n-1)
In Undirected Graph e=(n*(n-1))/2

9. Graph Terminology
Complete Graph:
A B
C
Directed Complete Graph
e= n*(n-1)= 3*2 = 6
Vertices V(G) = {A, B, C }
Edges E(G) = {(A,B), (A,C), (B,A), (B,C), (C,A),
(C,B)}

10. Graph Terminology
Complete Graph:
A B
C
Undirected Complete Graph
e= (n*(n-1))/2= 6/2=3
Vertices V(G) = {A, B, C }
Edges E(G) = {(A,B), (A,C), (B,C)}

11. Graph Terminology
Weighted Graph: A graph in
which every edge carries a
value.
A B C
D E
50
30
20
40
30 50

12. Graph Terminology
Loop: In graph loop (also called
a self-loop) is an edge that
connects a vertex to itself. A
simple graph contains no
loops..
A B C
D E
In this graph vertex C having
loop

13. Graph Terminology
Graph Degree:
Degree of the Vertex
In-Degree of the vertex
Out-Degree of the vertex
 In this Degree works on
undirected Graph
 In-Degree and Out-Degree works
on Directed Graph

14. Graph Terminology
Degree: the number of
relations a particular node /
Vertex makes with the other
node / Vertex in the graph.
d(v).
In this graph at vertex D out degree is d(D) =
3
A B C
D E

15. Graph Terminology
Out-Degree: if V is vertex of a
directed graph D number of
edges for which v is the initial
vertex is called out- degree. It is
denoted by d+
(v).
A B
D
In this graph at vertex A out degree is d+
(A) =
2

16. Graph Terminology
In-Degree: if V is vertex of a
directed graph D number of
edges for which v is the
terminal vertex is called in
degree. It is denoted by d-
(v).
A B
D
In this graph at vertex D in degree is d-
(D) = 1

17. Graph Terminology
Note1: in an undirected graph
Degree of total Graph is sum of
all the vertex degrees.
d(G)=∑ d(v)
Note2: in a directed graph
Degree of total Graph is sum of
in-degree and out-degree.
d(G)=∑ d+
(v) + ∑ d-
(v)

18. Graph Terminology
Note3: all vertex in-degree and
out-degree always same.
∑ d+
(v) = ∑ d-
(v)
Note4: in a graph Total number
of edges is equal to the half of
the tree degree.
e= ∑ d(v) / 2

19. Graph Terminology
Note5: in a Directed graph Total
number of edges is equal to all vertex
in-degree or out-degree.
e = ∑ d+
(v) = ∑ d-
(v)
Note6: Self loop can be treated as 1
outer and one inner so total 2.
d+
(v) = d-
(v) = 1
C

20. Graph Terminology
A d(A) = 2
vertex Degree
B d(B) = 3
C d(C) = 2
D d(D) = 3
E d(E) =
2
A B C
D E
Vertex (v)=5
edges (e)=6
e=∑ d(v)/2
∑ d(v)=2*e
∑ d(v)=2*6=12
∑ d(v)=2+3+2+3+2=12

21. Graph Terminology
A B C
D E
A d+
(A) = 2 d-
(A) = 0
vertex Out-Degree In-Degree
B d+
(B) = 1 d-
(B) = 2
C d+
(C) = 0 d-
(C) = 2
D d+
(D) = 2 d-
(D) = 1
E d+
(E) = 1 d-
(E) = 1
e=∑ d+
(v)=∑ d-
(v)
∑ d+
(v)=2+1+0+2+1=6 ∑ d-
(v)=0+2+2+1+1=6
Vertex (v)=5
edges (e)=6

22. Graph Terminology
A B C
D E
A d+
(A) = 2 d-
(A) = 0
vertex Out-Degree In-Degree
B d+
(B) = 1 d-
(B) = 2
C d+
(C) = 1 d-
(C) = 3
D d+
(D) = 2 d-
(D) = 1
E d+
(E) = 1 d-
(E) = 1
e=∑ d+
(v)=∑ d-
(v)
∑ d+
(v)=2+1+1+2+1=7 ∑ d-
(v)=0+2+3+1+1=7
Vertex (v)=5
edges (e)=7

23. Tree Vs Graph
In Tree only path allowed
between nodes but in Graph
nodes can have directional or
unidirectional.
In Tree each node having only
one parent but in Graph there is
no such concept.
In tree there is no concept loops
and self loops unlike graphs.

24. Tree Vs Graph
Tree uses traversed using in-
order, pre-order, post-order but
Graph uses BFS(Breadth First
Search) and DFS(Depth First
Search).
Tree is a hierarchical structure
and Graph has a network
model.