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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[1], Liu[2], Rosen[3] and others. A "simple path" according to another group (Cormen et al[4], Stanat and McAllister[5] 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),

Questions & Answers

what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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