GRAPH THEORY OF NUMBER THEOREM IN DISCRETE MATH

Contents
Introduction of
Graph
Types of Graph
Paths and Circuits
Graph Applications

GROUP
MEMBERS
Khadija Hanif(2024-DS-31)
Armeen Asim(2024-DS-44)
Faiza Ameen(2024-DS-20)
Tayyaba Amir(2024-DS-41)

A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, 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.
G = [V(G),E(G)]
GRAPH

Basic Terminologies
Adjacent Vertices
Two vertices a and b of a graph G(V,E) are said to be
adjacent if there is an edge = (a,b) connecting a and
b.
In given graph a and b , b and c, c and d, and b and d
are adjacent vertices.
Loop
An edge which is incident from and into itself (that
starts and ends at the same vertex). Such an edge is
called a loop.
In the given graph b forms a loop.

A directed graph (or digraph) (V, E) consists of a
nonempty set of vertices V and a set of directed edges
(or arcs) E. Each directed edge is associated with an
ordered pair of vertices. The directed edge associated
with the ordered pair (u, v) is said to start at u and
end at v.
Directed Graph

Simple Directed Graph
When a directed graph has no loops and has no multiple
directed edges, it is called a simple directed graph. Because a
simple directed graph has at most one edge associated to each
ordered pair of vertices (u, v), we call (u, v) an edge if there is an
edge associated to it in the graph.
Directed Multigraph
Directed graphs that may have multiple directed edges from a
vertex to a second (possibly the same) vertex are known as
directed multigraphs.

An undirected graph consists vertices V and edges (or
arcs) E, whose direction is not specified. The edges of
an undirected graph are also said to be undirected.
Undirected Graph

Mixed Graph
A graph with both directed and undirected
edges is called a mixed graph
Pseudograph
Graphs that may include loops, and possibly multiple
edges connecting the same pair of vertices or a vertex to
itself, are sometimes called pseudographs.

Answer
Given graph is;
• Undirected
• Yes, it has multiple edges
• Yes, It contains 3 loops
Question
Determine whether the graph shown has directed or
undirected edges, whether it has multiple edges, and
whether it has one or more loops.

Multigraph
Graphs that have more than one edge connecting the same pair
of vertices. Graphs that may have multiple edges connecting the
same vertices are called multigraphs.
Simple Graph
A graph in which each edge connects two different vertices and
where no two edges connect the same pair of vertices is called a
simple graph.

Finite Graph
A graph with a finite vertex set and a finite edge set is called a
finite graph.
Infinite Graph
A graph with an infinite vertex set or an infinite number of edges
is called an infinite graph.

Special Graphs
Noncomplete Graph
A simple graph for which there is at least one pair of distinct
vertex not connected by an edge is called noncomplete.
Complete Graph
A complete graph on n vertices, denoted by Kn, is a simple graph
that contains exactly one edge between each pair of distinct
vertices. The graphs Kn, for n = 1, 2, 3, 4, 5, 6, and so on.

Cycle Graph
A cycle Cn, n ≥ 3, consists of n vertices V1, V2, … , Vn and edges
{V1, V2}, {V2, V3}, … , {Vn−1, Vn}, and {Vn, V1}.
Wheel Graph
We obtain a wheel Wn when we add an additional vertex to a
cycle Cn, for n ≥ 3, and connect this new vertex to each of the n
vertices in Cn, by new edges.

n-Cubes
An n-dimensional hypercube, or n-cube, denoted by Qn, is a
graph that has vertices representing the 2n
bit strings of length n.
Two vertices are adjacent if and only if the bit strings that they
represent differ in exactly one bit position.
Q1 = 21
= 2 ……… (1, 0)
Q2 = 22
= 4 ……… (10, 00, 01, 11)

Question
How many vertices will be in Q3 and what would be the bit strings?
Q3 = 23
= 8 ……… (100, 110, 111, 101, 001, 000, 010, 011)
Answer

WALK
In graph theory, a walk is defined as:
• a sequence of vertices and edges
that alternates between them,
starting and ending at specified
vertices.
• A walk can traverse the same edge
or vertex multiple times.
a-e1-b-e2-c-e3-d
V-E-V-E-V-E-V
WALK

OPEN WALK
OPEN WALK CLOSED WALK
• The open walk starts at
one vertex and ends at a
different vertex. Thus, it
does not necessarily
return to the initial
vertex.
• A closed walk is defined
as a sequence of
vertices and edges in a
graph where The walk
starts and ends at the
same vertex.
• An open walk can visit
the same vertex or
traverse the same edge
multiple times.
• Like open walks, a closed
walk can traverse the
same edge or visit the
same vertex multiple
times.

LENGTH OF THE WALK
The length of a walk is defined as the total number of edges
that are included in the walk.
 In this example:
a-b-c-d-b-c-d is a walk
What will be the length of the walk?
• e1-e2-e3-e5-e2-e3 or
• one less than vertices - length of walk

TRAIL
CIRCUIT
• In graph theory, a trail is a specific type of walk with certain restrictions
on the repetition of edges.
• Open walk
• Vertex can be repeated
• In graph theory, a circuit (also known as a cycle) is a specific type of
• closed walk that starts and ends at the same vertex
• with the restriction that it does not repeat any edges or vertices (other
than the starting and ending vertex).

EXAMPLES
WHICH ONE OF THE
FOLLOWING IS TRAIL AND
WHICH ONE IS CIRCUIT?

PATH
• Path is a walk with no repeated vertex. Thus, no
edges will be repeated.
• It is a trail is which neither edges are repeated i.e.
if we transverse graph such that we do not repeat
the vertex nor we repeat an edge
• “open walk with no vertices and edge repeated”
Both are a path

CYCLE
• CLOSED PATH is a CYCLE
• Traversing a graph such that we do not repeat a
vertex nor an edge but the starting and ending vertex
can be same i.e. starting and terminal vertex can be
same only then we get a cycle.
• “ Closed walk with no edges and vertex repeating
except starting and last vertex”
Both are cycle

VERTEX REPEAT EDGE REPEAT ENDING AND
STARTING POINT
REPEAT
WALK
TRAIL
CIRCUIT
PATH
CYCLE
CLOSED-YES
OPEN-NO

UNDERSTANDING PATHS
IN GRAPH THEORY
 Many problems can be modeled using paths in
graphs.
Example applications include:
• Sending messages between computers.
• Routing for mail delivery.
• Planning garbage pickup.
• Analyzing diagnostics in computer networks.

Paths in Acquaintanceship Graphs
An acquaintanceship graph is defined as:
" A graph that models the relationships between people ,where Vertices
represent people and edges represent people who know each other.”
Example

“The acquaintanceship graph of all people in the world has more than six billion
vertices and probably more than one trillion edges!”
“Many social scientists have conjectured that almost every pair of people
in the world are linked by a small chain of people, perhaps containing just
five or fewer people. This would mean that almost every pair of vertices in the
acquaintanceship graph containing all people in the world is linked by a path of
length not exceeding four.”

Paths in collaboration graphs
Collaboration graph is defined as:
“A graph modeling some social network where the vertices represent participants of
that network (usually individual people) and where two distinct participants are
joined by an edge whenever there is a collaborative relationship between them of
a particular kind.”
Erdos Number
“The Erdős number describes the "collaborative distance “ between mathematician Paul
Erdős and another person, as measured by
authorship of mathematical papers.”
Collaboration graph of mathematicians also known as the Erdős collaboration graph is the
graph where two mathematicians are joined by an edge whenever they co-authored
a paper together.

This table refers to mathematicians and their collaborative distance from
the famous Hungarian mathematician Paul Erdős, who published
extensively in mathematics. The Erdős number measures how many steps
(collaborative links) a mathematician is away from having co-authored a
paper with Paul Erdős.
 Erdős Number 0: Paul Erdős himself (1 person).
 Erdős Number 1: Mathematicians who directly co-authored a paper
with Paul Erdős (504 people).
 Erdős Number 2: Mathematicians who co-authored a paper with
someone who has an Erdős number of 1, but not with Erdős directly
(6,593 people).
 The numbers grow as the Erdős number increases, showing how the
collaborative network expands as the distance increases.

Bacon Number
“In the Hollywood graph, the Bacon number of an actor
is defined to be the length of the shortest path connecting c
and the well-known actor Kevin Bacon.”
Collaboration graph of movie actors, also known as the Hollywood graph, where two movie
actors are joined by an edge whenever they appeared in a movie together.

This table refers to actors and their collaborative distance from Kevin
Bacon, a Hollywood actor. The Bacon number measures how many steps
(film collaborations) an actor is away from having worked on a movie
with Kevin Bacon.
• Bacon Number 0: Kevin Bacon himself (1 person).
• Bacon Number 1: Actors who directly worked in a film with Kevin
Bacon (3,452 people).
• Bacon Number 2: Actors who acted with someone having a Bacon
number of 1 but not directly with Kevin Bacon (401,636 people).
Similarly, as the Bacon number increases, the number of actors in that
category initially increases but then decreases as the collaborative
distance becomes larger.

Question
Can all the 9 vertices given below be
attached with exactly 5 other
vertices?

Answer
No, not every vertex can be linked with
exactly 5 vertices.
All the vertices are linked with exactly 5
vertices except for the last(9th
) one.
It has been linked with 4 other vertices
and cannot be
attached to the any other vertex
(theoretically).

Applications of Special Types of Graphs
Graphs help us model and solve real-world problems, particularly in computer networks and
parallel processing.
Local Area Networks (LANs)
A local area network connects computers and devices within a small
area (like a building). There are different designs for these networks,
and graphs are used to represent them
Star Topology
 Structure: All devices (like computers and printers) are connected
to one central device (a hub or a switch).
 Representation: This can be represented by a complete bipartite
graph K1,n.
 Function: Data travels through the central device to reach other
devices.
Star Topology Graph

Ring Topology
 Structure:
Each device is connected to exactly two other devices, forming a
circular data path.
 Representation:
This can be represented by an n-cycle graph, Cn.
 Function:
Data travels around the ring until it reaches the destination.
Ring Topology Graph

Hybrid Topology
 Structure:
Combines elements of both star and ring topologies.
 Representation:
This can be represented by a wheel graph, Wn.
 Function:
Data can travel through the ring or via the central device,
providing redundancy.
Hybrid Network

Parallel Computation and
Interconnection Networks
Parallel processing involves using multiple processors to solve computationally intensive problems more
efficiently. Different interconnection networks are used to connect these processors.
Complete Graph (Kn)
• Structure: Each processor is directly connected to every other processor.
• Representation: Kn graph.
• Function: Allows for maximum communication speed.
• Advantage: Fast communication.
• Disadvantage: Requires a lot of connections, which can be impractical for large numbers of
processors.

Linear Array
• Structure:
Processors are connected in a straight line.
• Representation:
Linear array graph.
• Function:
Each processor is connected to its immediate neighbors.
• Advantage:
Simple and requires fewer connections.
• Disadvantage:
Data may need to hop through many processors to reach its destination.

Mesh Network (Two-
Dimensional Array)
• Structure:
Processors are arranged in a grid, each connected to its four
immediate neighbors.
• Representation:
Mesh network graph.
• Function:
Provides multiple pathways for data.
• Advantage:
Balances connection cost and communication efficiency.
• Disadvantage:
Edge processors have fewer connections, leading to potential
bottlenecks.

Hypercube Network
• Structure:
Processors are connected in a structure that resembles a multi-dimensional cube.
• Representation:
Hypercube graph (Qn).
• Function:
Each processor is connected to others that differ by only one bit in their binary
representation.
• Advantage:
Good balance of connections and communication pathways.
• Disadvantage:
More complex to understand and implement.

GRAPH THEORY OF NUMBER THEOREM IN DISCRETE MATH

More Related Content

Similar to GRAPH THEORY OF NUMBER THEOREM IN DISCRETE MATH (20)

Recently uploaded (20)

GRAPH THEORY OF NUMBER THEOREM IN DISCRETE MATH