The document discusses graphs and graph theory. It defines graphs as non-linear data structures used to model networks and relationships. The key types of graphs are undirected graphs, where edges have no orientation, and directed graphs, where edges have orientation. Graph traversal algorithms like depth-first search and breadth-first search are discussed. Common graph terminology is defined, including vertices, edges, paths, cycles, degrees. Different graph representations like adjacency matrices and adjacency lists are also covered. Applications of graphs include modeling networks, routes, and relationships.

1. Graphs
By
Dr. P. Amudha
Associate Professor
School of Engineering
Avinashilingam Institute

2. Overview
 Graphs
◦ undirected graphs
◦ directed graphs
 Graph Traversals
◦ depth-first (recursive vs. stack)
◦ breadth-first (queue)

3. Introduction to graph
 Non-linear data structure used in
solving games and puzzles
 No root node as in trees
 Complex Data structure
 Graph Theory : study of graphs in CS

4. Contd.,
• Graphs are used to represent network
structures
Computer network Route map of Coimbatore
• Graphs unlike trees, do not represent
hierarchical relationship

5. Applications of graph
 Analysis of electrical circuits
 Finding shortest routes
 Project Planning
 Social sciences
 Cybernetics
 Linguistics ………..

6. What is a graph?
 A data structure that consists of a set of
nodes (vertices) and a set of edges that
relate the nodes to each other
 The set of edges describes relationships
among the vertices

7. Formal definition of graphs
 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)
 Where,
V = { v1,v2,v3,…vn}
E = {e1,e2,e3,…en}

8. 8
Examples of graphs
•Vertices drawn with circles
•Edges drawn with lines

9.  Trees are special cases of graphs!!
Trees vs graphs

10. Types of Graphs
Basically two types:
 Directed Graphs
 Undirected graphs

11. 11 Lecture 7: Graphs
Directed Graphs
 A directed graph is a graph which
consists of directed edges, where
each edge in E is unidirectional.
 Also referred as digraph.
 Each edge connects two vertices,
called the source and target
 If (v,u) is a directed edge, then (v,u) 
(u,v)

12. 12 Lecture 7: Graphs
Directed Graphs
 A directed graph is a finite set of
vertices and a finite set of edges.
 Each edge connects two vertices,
called the source and target; the edge
connects the source to the target
(order is significant)

13. 13 Lecture 7: Graphs
Directed Graphs
V1
V3
V2
Useful for modelling directional
phenomena, like geographical
travel, or game-playing where
moves are irreversible
Examples: modelling states in a
chess game, or Tic-Tac-Toe
Arrows are used to represent
the edges; arrows start at the
source vertex and point to the
target vertex
(V1,V2)  (V2,V1)

14. Example:
Directed graphs (cont.)
E(Graph2) = {(1,3), (3,1), (11,1), (5,9), (9,11), (9,9),
(5,7)

15. Undirected Graphs
 An undirected graph is a graph, which
consists of undirected edges.
 If (v,u) is an undirected edge then
(v,u)=(u,v)
 Each edge connects two vertices
 The order of the connection is
unimportant
V1 V2
V3
(V1,V2) = (V2,V1)

16. L23 16
Simple Graphs
Different purposes require different types
of graphs.
Eg: Suppose a local computer network
◦ Is bidirectional (undirected)
◦ Has no loops (no “self-communication”)
◦ Has unique connections between computers

17. L23 17
Multigraphs
 If computers are connected via
internet, there may be several routes
to choose from each connection.
 Depending on traffic, one route could
be better than another.
 It allows multiple edges, but still no
self-loops

18. 0 2
1
(a)
2
1
0
3
(b)
Example of a graph with feedback loops and a multigraph
self edge multigraph:
multiple occurrences
of the same edge

19. Graph terminologies
 Adjacent Nodes/ Adjacent vertex
 Path
 Length
 Cycle
 Degree
 Weighted graph
 Complete graph
 Subgraph
 Strongly Connected graph

20. Adjacent nodes and path
 Two nodes are adjacent if they are connected
by an edge
 A path in a graph is a sequence of vertices,
each adjacent to the next
A D
EB
FC
adjacent (A) ={B,C,D}
adjacent (B) ={A,C,E}
adjacent (C) ={A,B,D,E}
adjacent (D) ={A,C,E,F}
adjacent (E) ={B,C,D,F}
adjacent (F) ={E,D}
Path 1: A -> B -> E -> F
Path 2: A -> C -> E -> F
Path 3: A -> D -> F
Path 4: A -> C -> D -> F

21. Length of the path and Cycle
 Length of the path is the number of edges
on the path, which is equal to N-1, where N
is the number of vertices.
 Cycle in a graph is a path in which first and
last vertices are the same.
V1 V2 V3 V1
 A graph which has cycles is referred to as
cyclic graph
V3 V2
V1 Length of the path V1 to V2
is 2.
ie., (V1,V2), (V2,V3)

22. Degree
 The number of edges incident on a
vertex V determines its degree.
 In undirected graph, degree(V) = number
of edges incident at V.
 In directed graph, it is categorized into
indegree and outdegree.
 Indegree (V) = no. of edges entering into
the vertex V
 outdegree (V) = no. of edges exiting from
the vertex V

23. Contd.,
Vertex Degree
A
B
C
D
3
2
2
3
A B
C D
Table 1:Degree of each vertex of an undirected graph
A
DC
B Vertex Indegree Outdegree
A
B
C
D
2
1
2
1
1
2
1
2
Table 2: indegree and outdegree of each vertex of directed graph

24. Acyclic graph
 A directed graph which has no cycle is
referred to as acyclic graph.
 It is abbreviated as DAG (Directed
Acyclic Graph)
A
B C
D
E

25. Weighted Graph
 A graph is said to be weighted graph if every edge
in the graph is assigned some weight or value.
 The weight of the edge is a positive value the may
be representing the cost or distance between the
vertices.
 For example, in a road map represented as a
graph, weight of the edges can be based on the
distance between the cities or functions of distance
and traffic in the road

26. Complete and Incomplete Graph
0
1 2
3
0
1 2
3 4 5 6
G1
G2
V(G1)={0,1,2,3} E(G1)={(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)}
V(G2)={0,1,2,3,4,5,6} E(G2)={(0,1),(0,2),(1,3),(1,4),(2,5),(2,6)}
complete graph incomplete graph
A complete graph is a graph in which every vertex is
directly connected to every other vertex

27.  What is the number of edges in a
complete directed graph with N
vertices?
N * (N-1)
Here in this example,
N = 4
No. of edges = 4 * 3
= 12
Contd..

28.  What is the number of edges in a complete
undirected graph with N vertices?
N * (N-1) / 2
Here in this example,
N = 5
No. of edges = 5 * 4 / 2
= 10
Contd.,

29. Subgraph
 A subgraph of G is a graph G’ such
that V(G’) is a subset of V(G) and E(G’)
is a subset of E(G)

30. 0 0
1 2 3
1 2 0
1 2
3
(i) (ii) (iii) (iv)
(a) Some of the subgraph of G1
0 0
1
0
1
2
0
1
2
(i) (ii) (iii) (iv)
(b) Some of the subgraph of G2
0
1 2
3
G1
0
1
2
G2
Subgraphs of G1 and G2

31. Connected Graph
 A graph is called connected if there is
a path from any vertex to any other
vertex.
A B
C D
Connected Graph
A
C D
B
Unconnected Graph

32. Graph representation
 A graph is a mathematical structure
and it must be represented in some
kind of data structure.
 Two ways to represent the graph:
 Adjacency matrix
 Adjacency list

33. Array representation
 Simple way to represent a graph is to
use 2D array.
 This is known as adjacency matrix
representation.
Adjacency matrix A for a graph G =(V,E)
with n vertices is an n x n matrix such
that,
Aij = 1, if there is an edge Vi to Vj
Aij = 0, if there is no edge

34. Contd.,
 Adjacency matrix for undirected graph
 Adjacency matrix for directed graph
V1
V4V3
V2 V1 V2 V3 V4
V1 0 1 1 0
V2 1 0 1 1
V3 1 1 0 1
V4 0 1 1 0
V1
V4V3
V2 V1 V2 V3 V4
V1 0 1 1 0
V2 0 0 0 1
V3 0 1 0 0
V4 0 0 1 0

35. Contd.,
 Adjacency matrix for weighted graph
 Aij = Cij if there exists an edge from Vi to Vj
 Aij = 0 if there is no edge and i = j
 If there is no edge for i to j, assume C[i,j] =
V1
V4V3
V2
V1 V2 V3 V4
V1 0 3 9 
V2  0  7
V3  1 0 
V4  1 8 0
3
7
8
9 1

36. Adjacency list representation
 In this representation, graph is stored
as linked structure.
 Linked list representation of undirected graph:
A
DC
B
A
B
C
D
B
A
C
A
C Null
C
B
B Null
D Null
D Null

37. Contd.,
 Linked list representation of directed graph:
A
DC
B
A
B
C
D
B
D Null
C Null
A Null
D Null

38. Topological Sort
 A topological sort is a linear ordering of vertices in a
directed acyclic graph such that if there is a path from
Vi to Vj, then Vj appears after Vi in the linear ordering.
 For example, a directed edge between (C1,C2)
indicates that C1 must be completed before
attempting to course C2.
10th
Arts
Eng
g
Med
PG
12th
Dip

39. Topological Order Algorithm
Algorithm Topological order (Graph G)
1. Find any vertex with no incoming
edges.
2. If such vertex is found, remove it
along with its edges from the graph.
3. Repeat step 1 for all the vertices.
END Topological order;

40. Implementing topological sort
1. Find the indegree for every vertex.
2. Enqueue the vertices whose indegree is ‘0’ on
the empty queue.
3. Dequeue the vertex V and decrement the
indegree’s of all its adjacent vertices.
4. Enqueue the vertex on the queue, if the indegree
falls to ‘0’.
5. Repeat from step 3 until the queue becomes
empty.
6. The topological ordering is the order in which the
vertices are dequeued.

41. Example
a b c d
a 0 1 1 0
b 0 0 1 1
c 0 0 0 1
d 0 0 0 0
a
cb
d
Adjacency matrix
The figure states that a precedes b, c; b precedes c, d and so on.
Step 1: Number of 1’s present in each column of adjacency matrix
represents the indegree of the corresponding vertex.
(ie.,) indegree(a) = 0 indegree(b) = 1
indegree(c) = 2 indegree(d) = 2
Step 2: Enqueue the vertex whose indegree is ‘0’.
vertex a is ‘0’, so place it on the queue.

42. Graph implementation
 Array-based implementation
◦ A 1D array is used to represent the vertices
◦ A 2D array (adjacency matrix) is used to
represent the edges

43. Array-based implementation

44. Graph implementation (cont.)
 Linked-list implementation
◦ A 1D array is used to represent the vertices
◦ A list is used for each vertex v which contains
the vertices which are adjacent from v
(adjacency list)

45. Linked-list implementation