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Proof

The Euler characteristic for lines is 1, which means that # vertices - # edges = 1 . Therefore there are n + k - 1 edges in a tree, where n is the number of terminal points and k is the number of Steiner points.

Each Steiner point has 3 edges coming from it, and each terminal point has at least one, so there are at least 3 k + n 2 total edges; the division by 2 accounts for the fact that each edge is counted at two vertices. Therefore

n + k - 1 3 k + n 2

which, by simple algebra, yields

n - 2 k .

Definition

A Steiner tree with n - 2 Steiner points is a full Steiner tree.

Theorem 5

Every Steiner tree that is not a full Steiner tree can be decomposed into a union of full Steiner trees.

Proof

See [link]

Examples

In a 3 by 4 rectangle, two Steiner trees can be created, but the one with edge between the Steiner points parallel to the long side is the Steiner minimal tree.

In a 1 by 2 rectangle, the minimum spanning tree is not a Steiner tree.

Definition

A topology on a graph determines how many terminals and Steiner points there are, and how the points are connected to each other. A topology is a degeneracy of another if it can be obtained from shrinking edges of the original. The set of degeneracies of a topology G is denoted by D ( G ) .

A topology is a Steiner topology (full Steiner topology) if every Steiner point has degree 3 (and terminal has degree 1).

If F is a full Steiner topology, then D s ( F ) is the set of Steiner topologies in D ( F ) .

It is easy to see that a Steiner tree contains no closed shapes, called cycles. Otherwise, one edge could be deleted, and the resulting structure would be of shorter total length while still connecting all of the points.

Theorem 5

A Steiner tree whose topology is in D ( F ) for some full Steiner topology F is the unique minimum tree among trees whose topologies are in D s ( F ) .

Proof

Let T be a tree whose topology G D s ( F ) . Let v 1 , ... , v n be an enumeration of the vertices. Define f i j = 1 if [ v i , v j ] is an edge in G and f i j = 0 otherwise. Then the length of T is

T = i < j f i j v i - v j

| v i - v j | is a norm, and norms are convex due to the triangle inequality, so T is strictly convex except when edges remain the same direction in a perturbation. In that case, when the three edges from a Steiner point are parallel, moving the Steiner point towards the side of the two overlapping edges decreases the total length, and thus | T | is strictly convex. Because a Steiner tree is a local minimum, strict convexity guarantees that it is the unique minimum among trees with topologies in D s ( F ) .

Corollary

There exists at most one relatively minimal tree for a given topology.

Definition

A Steiner hull for a given set of points N is a region which is known to contain a Steiner minimal tree.

In the Euclidean plane, the Steiner hull is bounded.

Lemma

Let [ v i , v j ] be any path in a Steiner minimal tree, where v i and v j are defined as before. Let L ( v i , v j ) be the region consisting of all points p satisfying

| p - v i | < | v i - v j | and | p - v j | < | v i - v j |

L ( v i , v j ) is the lune-shaped intersection of circles of radius | v i - v j | centered on v i and v j . No other vertex of the Steiner minimal tree can lie in L ( v i , v j ) .

Questions & Answers

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
what school?
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?
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absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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it is a goid question and i want to know the answer as well
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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 ?
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for screen printed electrodes ?
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What is lattice structure?
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Michell trusses study, rice u. nsf vigre group, summer 2013. OpenStax CNX. Sep 02, 2013 Download for free at http://cnx.org/content/col11567/1.2
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