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Vizing's Conjecture is a lower bound for the domination number of the Cartesian Product of two graphs in terms of the domination number of the separate graphs. This module addresses observations about certain graph properties that can be assumed for Vizing's Conjecture and a related conjecture on independent domination numbers.

Introduction

Preliminary definitions

Graph

A graph is a set G ( V , E ) with V a set of vertices and E a set of edges or vertex pairs. Two vertices v 1 , v 2 V are adjacent if the vertex pair ( v 1 , v 2 ) are in E . Graphs are a common model for networks.

A graph

Complete graph

A graph G on n vertices is a complete graph if for each pair v 1 , v 2 V ( v 1 , v 2 ) E . Call K n the complete graph on n vertices.

Complete graph on 4 vertices

Cartesian product graph

Given graphs G and H the Cartesian Product Graph is defined to be G H with

V ( G H ) = { ( v , w ) : v G , w H } E ( G H ) = { ( ( v 1 , w 1 ) , ( v 2 , w 2 ) ) : v 1 = v 2 and ( w 1 , w 2 ) E ( H ) or w 1 = w 2 and ( v 1 , v 2 ) E ( G ) }

The cartesian product of 2 complete graphs makes a "cheese block"

Neighbors

Given a graph G ( V , E ) and a set S V then we define the neighbors of S to be the set

N ( S ) = { v : v V and ( v , s ) E for some s S } S

and similarly the closed neighborhood is the set

N [ S ] = { v : v V and ( v , s ) E for some s S } S

Dominating set

Given a graph G ( V , E ) , a set D V is a dominating set if N [ D ] = V .

A star graph showing 2 dominating sets (red and cyan)

Domination number

Given a graph G ( V , E ) , the domination number of G is

γ ( G ) = min { | D | : N [ D ] = V }

K-critical graph

A graph G ( V , E ) , is called k-edge-critical (or k-critical , for short) if γ ( G ) = k , and, u , v V ( G ) such that u and v are not adjacent, γ ( G + u v ) < k .

Independent set

Given a graph G ( V , E ) , a set I V is independent if for all v , w I ( v , w ) E . An independent set is maximal if it is not a subset of any other independent set.

A maximal independent set (cyan)

Independence number

Given a graph G ( V , E ) , the independence number denoted i ( G ) is defined by

i ( G ) = m i n ( { | I | : I is a maximal independent set } )

Domination theory

Domination Theory is an emerging field in Graph Theory addressing how to find dominating sets for certain graphs and important models in the theory.

Applications

Domination Theory is a very interesting subfield of graph theory because it has many real-world applications. Finding a minimum set whose closed neighborhood encompasses a network has obvious implications for minimum-cost ways of altering a network, or cheaply distributing goods throughout a network. For example, dominating set theory can help cell phone companies place a minimum number of towers to insure coverage for all of its clients. Similarly, dominating set theory can be useful for social marketing, in order to succesfully spread news about a product by using a minimal number of advertisements. In a less business-minded view, domination theory can help modeling the squares which are connected by the moves of a chess piece (such as the queen). This can be useful for solving problems like the maximum-placement problem, for an arbitary chess board, or for pieces with different movements. Lastly, domination theory can also have applications in facility location problems, such as finding the minimum distance to travel to one out of a set of locations (such as a police station). [link]

Questions & Answers

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Damian Reply
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Daniel
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s. Reply
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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.
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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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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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.
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s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
what is system testing
AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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Prasenjit
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Damian
silver nanoparticles could handle the job?
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Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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