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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
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
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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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