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PRESIDENCY
COLLEGE
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Topic: Types of graph, construction of graphs ,
theorems ...
Date of presentation: 16/09/22
TEACHER : MS. SAVITHA GOWDA MAM
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A graph consists of a nonempty set of vertices or
nodes and is a set of edges.
Each edge has either one or two vertices associated
with it called its endpoints. An edge is said to connect
its endpoints.
INTRODUCTION TO GRAPHS
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BY
RISHI RAJ
DIRECTED GRAPH AND
UNDIRECTED GRAPHS AND ITS
CONSTRUCTION
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A directed graph, also called a digraph, is a graph in
which the edges have a direction. This is usually
indicated with an arrow on the edge
DIRECTED GRAPHS
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• A directed graph (or digraph) is a set of vertices
connected by edges, where the edges have a
direction associated with them.
• In a directed graph a given set of vertices can have
multiple edges connected to them
• The construction of directed graph consists of
edges having direction represented by an arrow
mark pointing towards the respected vertex
Construction Of Directed Graph
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DIRECTED GRAPH ANOTHER EXAMPLE
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An undirected graph is a graph, i.e., a set of objects
(called vertices or nodes) that are connected, where
all the edges are bidirectional.
UNDIRECTED GRAPH
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UNDIRECTED GRAPH ANOTHER EXAMPLE
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In the above figure we find that :
1. In the above graph there is no directed edges
2. As the definition of undirected graph states that
undirected graph is graph whose edges are not
directed , we can conclude that the above figure is
an undirected graph.
3. The construction of the undirected graph consists
of five vertices connected with edges which are
not directed
Construction Of Undirected Graph
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BY
ROHESH KANCHAN
MULTIGRAPH AND
PSEUDOGRAPHS AND ITS
CONSTRUCTIONS
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Graphs that may have multiple edges connecting the
same vertices are multigraphs.
MULTI GRAPH
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CONSTRUCTION OF
MULTIGRAPH
Eg.1 In the figure we find that :
1. There are six vertices present in
the figure with edges connecting
their respective vertices.
2. On vertices B and C we can
observe that there are two
edges connecting the same set
of vertices.
3. By the definition of the
multigraph we can say that the
above figure is a multi-graph.
4. The construction of the
multigraph consisted of a set of
vertices and edges where
vertices B and C had two edges
connected to each other.
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CONSTRUCTION
OF MULTIGRAPH
Eg.2 In this graph
• There are 6 edges and 4
vertices or nodes i.e A1, A2,
A3, and A4.
• It has an edge connecting A1
and A4.
• According to the definition
we can get to know it is a
multigraph.
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GRAPHS THAT MAY INCLUDE LOOPS, AND POSSIBLY
MULTIPLE EDGES CONNECTING THE SOME PAIR OF
VERTICES OR A VERTEX TO ITSELF, ARE CALLED
PSEUDOGRAPHS.
PSEUDOGRAPHS
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CONSTRUCTION OF
PSEUDOGRAPH
In the figure we find that :
1. There are both graph loops
and multiple edges present in
the above graph.
2. As the definition of a pseudo
graph states that both graph
loop and multiple edges are
permitted in the pseudo
graph, we can conclude that
the above graph is a pseudo
graph.
3. The construction of the
pseudo graph consists of
edges, multiple edges,
vertices, and graph loops
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CONSTRUCTION OF
PSEUDOGRAPH
In this graph
• There are 4 nodes a, b, c,
and d. There is also a loop
present at C.
• According to the
definition, we can
conclude that it is a
Pseudograph.
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CYCLIC AND ACYCLIC GRAPHS
AND ITS CONSTRUCTIONS.
BY
LOHITH S
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• A graph with 'n' vertices (where, n>=3) and 'n'
edges forming a cycle of 'n' with all its edges is
known as cycle graph.
• A graph containing at least one cycle in it is known
as a cyclic graph.
• In the cycle graph, degree of each vertex is 2.
• The cycle graph which has n vertices is denoted by
Cn.
Cyclic Graph
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Example-1:
Construction Of Cyclic Graph
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• Example-2: The graph below is a cyclic graph
containing two cycles.
Construction Of Cyclic Graph
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• A graph which does not contain any cycle in it is
called as an acyclic graph.
• An acyclic graph does not contain cycles which
means , vertices V1,V2,V3…Vn ,edges
{V1,V2},{V2,V3}….{Vn,V1} does not hold true in the
following graph
• At some point in the graph the cycle breaks or
vertices are not connected with edges which ends
up breaking the cycle
Acyclic Graph
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• The following is an example for acyclic graph.
Construction Of Acyclic Graph
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Is this graph cyclic or acyclic?
Question
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BY
HARSHITH NAAG
CONNECTED AND
DISCONNECTED GRAPHS AND
ITS CONSTRUCTIONS.
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• A connected graph is a graph in which we can visit
from any one vertex to any other vertex.
• In a connected graph, at least one edge or path
exists between every pair of vertices.
• In a connected graph we can traverse from any one
vertex to any other vertex.
Connected Graph
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• Example: the below graph is an example for a
connected graph.
Construction of Connected Graph
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• A disconnected graph is a graph in which any path
does not exist between every pair of vertices.
• Disconnected pair of vertices are considered to be
a disconnected graph where no path exits between
the 2 pair of vertices
• two independent components in this case being a
pair of vertices ,which are disconnected are
considered to be disconnected graph
Disconnected Graph
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• Example: the below graph is an example for a
disconnected graph.
Construction of Disconnected Graph
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(Autonomous) BIPARTITE GRAPH AND COMPLETE
BIPARTITE GRAPH
By
Syed Mansoor
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If a set of vertices is divide into two disjoined sets V1 And V2 each edge
in the graph joins each vertex of V1 and V2. Then the graph is called
Bipartite Graph.
What is Bipartite Graph?
A
B D
C
V1 V2
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What is a Complete Bipartite Graph?
• Complete Bipartite graph is a graph whose vertices can be
divided into two disjoint and independent sets U and V, that is
every edge connects a vertex in U to one in V.
A
B
C
D
P
Q
R
U={A,B,C,D} V={P,Q,R}
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BY
SHUBHA
AND RISHANTH LAMA
THEOREMS
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Theorem 3 : Let G = (V,E) be a directed graph with
edge e then:
i.e, the sum of outdegree of the vertices of a diagraph
G equals to the sum of indegrees of the vertices which
equals the number of edges in G .
Theorems : Directed Graph
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Sum of indegree:
1+2+2+3= 8
Sum of outdegree:
2+4+1+1=8
The total number of
Edges= 8
Example 1:
Vertices A B C D
In
Degree 1 2 2 3
Out
Degree 2 4 1 1
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Sum of indegree :
0+1+3+2+4= 10
Sum of Outdegree:
3+2+2+2+1 = 10
The total number of edges= 10
EXAMPLE 2:
Vertices 1 2 3 4 5
Indegree 0 1 3 2 4
Outdegree 3 2 2 2 1
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Theorem 4: In a simple graph atleast 2 vertices will have
same degree. (n >= 2).
Theorem
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Theorem 5: In regular graph with n vertices has nk/2
edges.
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Find the total number of edges present in this
graph.
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Theorem 6: A graph (G) is bipartite If and only if it
contains no odd cycles .
Bipartite means the Vertices of graph are partitioned into sets V1 and V2 such
that no two verties of the same partition can be adjacent .
Odd length and even length cycles:
Theorem : Bipartite graph
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It's an even length cycle as
the number of edges=4 i.e,
a even number.
We are taking two sets
named V1 and V2.
V1= { x1, x2 }
V2={ y1, y2 }.
V(G)= V1 U V2
EXAMPLE 1:
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Here we take an odd cycle.
From the diagram,
1 belongs to V1
2 belongs to V2
n belongs to V1 as it's odd.
But 5 is again connected to 1 which is not
possible as the vertices in same set
should not be adjacent .
Therefore a graph is said to be
bipartite if and only if there is
no odd cycles.
EXAMPLE 2:
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If G is a graph with n vertices,
where n≥3 and deg(v)≥n/2, for
every vertex v of G then G is
Hamiltonian
Dirac’s Theorem
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n=4(n>=3)
n/2=4/2
DEG(A)=2
DEG(B)=2
DEG(C)=2
DEG(D)=2
Example
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If graph G is graph of order n≥3
such that for all distinct non
adjacent pairs of vertices u and
v, deg(u)+deg(v)≥n, then G is
Hamiltonian
Ore’s theorem
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n=4(4>=3)
a pair of non-adjacent vertices
(U,V)=(A,D)or(B,C)
deg(A)+deg(D)>=n
2+2>=4
4>=4
Example
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THANK YOU
RISHI RAJ
ROHESH KANCHAN
HARSHITH NAAG
LOHITH S
SHUBHA HARINI
RISHANTH LAMA
