Data Structure DSL- Lec #16 - Graph.pptx

Lecture #16
Course: Data Structure
Instructor: Mohammad Dawood Saddiqi
Date: Wednesday, August 13, 2025
Graph
In the name of Allah
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Outline
• Introduction
• Graph
• Uses of graph
• Type of Graph
• Basic Concepts
• Parent Node
• Loop edge
• Multiple Edges
• Path & Length of graph
• Graph Representation
Wednesday, August 13, 2025
Instructor: Mohammad Dawood Saddiqi - Ghalib
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Introduction
• Tree data structures represent one to many relationships.
• In real life we frequently come across problems that can be best
described by many to many relationships.
• Such problems cannot be solved using trees or other data structures.
• To deal with such problems, graph data structures are used.
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Graph
• Graph is a non-linear data structure. It is made up of sets of nodes and
lines.
• Nodes are called vertices or points.
• Lines are called edges or arcs.
• Lines are used to connect nodes.
• An edge of the graph is represented as:
e = [u, v]
• 'u' and 'v' denote the start and end nodes of edge 'e', respectively.
• They are also known as the tail and head nodes of edge
• Graphs can he used to represent essentially any relationship.
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Uses of graph
• Graphs can be used to represent essentially any relationship.
They are used to study problems in a wide variety of areas
including computer science, electrical engineering, chemistry,
etc.
• For example. The transport networks, the following graph
represents transportation links between four cites. The vertices
represent cities. The edges correspond to the roads that link the
cities.
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Herat
Kandahar
Kabul
Jalalabad

Type of Graph
• There are different types of graph some of them are mentioned
here.
1. Directed graph
2. Undirected graph
3. Weighted graph
4. Complete graph
5. Regular graph
6. Isomorphic graphs
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(1)- Directed Graph
• A directed graph has edges that have direction. A directed graph
is also called a digraph. E.g.
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A B
D E
C

(2)- Undirected graph
• An undirected graph has edges that have no direction. An
undirected graph is also called an undigraph.
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A B
D E
C

(3)- Weighted graph
• A graph which has a weight or number associated with each
edge is called weighted graph. Weighted of an edge is sometime
called its cost.
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B
A
C
20
08
15

(4)- Complete Graph
• A graph is said to be complete graph if there is an edge between every
pair of vertices.
• or A graph is said to be complete one, if each node of the graph is
adjacent with every other node.
• If a node has n nodes, then each node has (n-1) total degree. i.e. it is
connected with (n-1) nodes, so the number of edges can be calculated
by the formula e=n(n-1)/2; where n is the no of nodes.
• e= n(n-1)/2
• e= 4(4-1)/2
• e=12/2
• e= 6
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A B
C D

(5)- Regular Graph
• A graph in which each node has equal total degree is called regular
graph.
Total- degree: The degree of a vertex is the number of edges incident to
that vertex
For Directed Graph
In-degree: the in-degree of a vertex v is the number of edges that have v as the head
Out –degree: the out-degree of a vertex v is the number of edges that have v as the
tail
About vertices ‘A’
Total Degree: 4
In-Degree:2
Out-Degree:2
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B
A
C

(6)- Isomorphic Graph
• When two graphs have same behavior in term of graphical
properties are called isomorphic graph.
• Condition for isomorphism.
• Total number of nodes in two graphs must be same.
• Total number of edges in two graphs must be same.
• Same number of in-degree and out-degree of the two graphs
• Direction of edges must be same.
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E
D
C
A
B
C
B
A
D
E

Basic Concepts:
• Source & Sink Nodes: The node that has a positive out-degree
but zero in-degree is called source node.
The node that that has zero out-degree but positive in-degree is
called sink node.
‘A’ is source node.
‘C’ is sink node.
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B
A
C
20
08
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Parent Node
• A node is called parent node if it has total degree of one. In below
graph A is parent node
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A
B
C
D
E

Loop edge
• An edge is said to be a loop edge if the same node is its tail and
head. E.g.
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B
A
C

Multiple Edges
• A graph is said to be have multiple edges if it has more than one
edge have the same tail and head nodes.
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B
A
C

Path & Length of graph
• A list of nodes of a graph where each node has an edge from it to
the next node is called path. It is written as a sequence of nodes,
A,B,C.
• A path which repeats no Node, is known as simple path.
• A path which begins and end at same node is called cycle.
• Moving from one node to other node is called walk.
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Graph Representation
• There are two ways to represent a graph in computer memory.
1. Adjacency matrix
2. Adjacency list
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Adjacency Matrix
• An adjacency matrix is a table, represents the edges between vertices of
the graph. In case of weight graph the adjacency matrix represent the
weights of the edges.
• If an edge e=(u,v) exists b/w two vertices u and v, it is represented in the
matrix by 1. absence of an edge is represented by 0.
• The adjacency matrix consist only 0’s and 1’s.
• If there are N vertices in the graph, the adjacency matrix will be an N x
N matrix.
• The columns of the matrix represent the “to” end of edge, while rows of
the matrix represent “from” start of edge.
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Example
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A B C D
A 0 1 0 1
B 0 1 0 1
C 0 1 0 0
D 1 0 1 0
A
D
B
C

Adjacency List
• In this method the graph is represented in computer memory by two link
list. One Node list and second Adjacency List.
• In Node List we will store the Nodes while in Adjacency List we will
store the adjacent nodes of a node.
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Example
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A D
E
C
B

Summary
• Tree data structures represent one to many relationships.
• In real life we frequently come across problems that can be best
described by many to many relationships.
• Graph is a non-linear data structure. It is made up of sets of nodes
and lines.
• There are different types of graph some of them are mentioned
here. Directed graph, Undirected graph, Weighted graph,
Complete graph, Regular graph, Isomorphic graphs.
• There are two ways to represent a graph in computer memory.
1. Adjacency matrix
2. Adjacency list
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The End
Thank You
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Data Structure DSL- Lec #16 - Graph.pptx

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