# 0.6 Discrete structures relation  (Page 4/13)

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For example Figure 1 is a digraph with 3 vertices and 4 arcs.

In this figure the vertices are labeled with numbers 1, 2, and 3.

Mathematically a digraph is defined as follows.

Definition (digraph): A digraph is an ordered pair of sets G = (V, A), where V is a set of vertices and A is a set of ordered pairs (called arcs) of vertices of V.

In the example, G1 , given above, V = { 1, 2, 3 } , and A = {<1, 1>,<1, 2>,<1, 3>,<2, 3>} .

## Digraph representation of binary relations

A binary relation on a set can be represented by a digraph.

Let R be a binary relation on a set A, that is R is a subset of A×A. Then the digraph, call it G, representing R can be constructed as follows:

1. The vertices of the digraph G are the elements of A, and

2.<x, y>is an arc of G from vertex x to vertex y if and only if<x, y>is in R.

Example: The less than relation R on the set of integers A = {1, 2, 3, 4} is the set {<1, 2>,<1, 3>,<1, 4>,<2, 3>,<2, 4>,<3, 4>} and it can be represented by the digraph in Figure 2.

Let us now define some of the basic concepts on digraphs.

Definition (loop): An arc from a vertex to itself such as<1, 1>, is called a loop (or self-loop)

Definition (degree of vertex): The in-degree of a vertex is the number of arcs coming to the vertex, and the out-degree is the number of arcs going out of the vertex.

For example, the in-degree of vertex 2 in the digraph G2 shown above is 1, and the out-degree is 2.

Definition (path): A path from a vertex x0 to a vertex xn in a digraph G = (V, A) is a sequence of vertices x0, x1, ....., xn that satisfies the following:

for each i,  0 ≤i ≤ n - 1 ,  <xi , xi + 1>∈A , or  <xi + 1 , xi>∈A ,   that is, between any pair of vertices there is an arc connecting them. x0 is the initial vertex and xn is the terminal vertex of the path.

A path is called a directed path   if  <xi, xi + 1>∈A ,   for every i,   0 ≤i ≤n -1.

If the initial and the terminal vertices of a path are the same, that is,   x0 = xn,   then the path is called a cycle.

If no arcs appear more than once in a path, the path is called a simple path. A path is called elementary if no vertices appear more than once in it except for the initial and terminal vertices of a cycle. In a simple cycle one vertex appears twice in the sequence: once as the initial vertex and once as the terminal vertex.

Note: There are two different definitions for "simple path". Here we follow the definition of Berge, Liu, Rosen and others. A "simple path" according to another group (Cormen et al, Stanat and McAllister and others) is a path in which no vertices appear more than once.

Definition(connected graph): A digraph is said to be connected if there is a path between every pair of its vertices.

Example: In the digraph G3 given in Figure 3,

1, 2, 5 is a simple and elementary path but not directed,

1, 2, 2, 5 is a simple path but neither directed nor elementary.

1, 2, 4, 5 is a simple elementary directed path,

1, 2, 4, 5, 2, 4, 5 is a directed path but not simple (hence not elementary),

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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