Transcript Current Uses of Cryptography

Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Application of Graph Theory and
Ecosystems
Discrete Methods Group Project 2007
Erika Mizelle, Kaiem L. Frink,
Elizabeth City State University
1704 Weeksville Road
Elizabeth City, North Carolina 27909
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Abstract
The word graph (graf) comes from the Greek
word graphein and is a noun. It is a diagram
indicating any sort of relationship between
two or more things by means of a system of
dots, curves, bars, or lines. The word
ecosystem (e’ko sis’tem) is from the Greek
word oikos meaning habitat + system. It is
defined as a community of organisms and
their nonliving environment.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Introduction of Graph Theory
Applications of graph theory are primarily, but
not exclusively, concerned with labeled
graphs and various specializations of these.
Graphs can be used in almost any field of
study for various different reasons. This paper
will discuss how graph theory and its
applications can be used in ecosystems and
DNA sequencing.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Leonhard Euler (1707-1783)
Leonhard Euler Theory
Notable Individual
Leonhard Euler (1707-1783) Leonhard Euler was the son of a
Calvinist minister from the vicinity of Basel, Switzerland. At 13
he entered the University of Basel, pursing a career in theology,
as his father wished. At the University of Basel, Johann Bernoulli
of the famous Bernoulli family of mathematicians tutored Euler.
His interest and skills led him to abandon his theological studies
and take up mathematics. Euler obtained his masters degree in
philosophy at the age of 16. In 1727 Peter the Great invited him
to join the Academy at St. Petersburg. In 1736, Euler solved a
problem known as the Seven Bridges of Konigsberg. In 1741 he
moved to the Berlin Academy, where he stayed until 1766. He
then returned to St. Petersburg, where he remained for the rest
of his life.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Examples of Graphs

Figure 1.1
Simple Graph
Figure 1.2
Directed Graph
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Transportation networks
Transportation networks. The map of a
bus line route forms a graph. The nodes
(vertices) could represent the different
cities or states that the bus visits.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Communication network.
Communication network. A collection of
computers that are connected via a
communication network can be naturally
modeled as a graph in a few different
ways. First, we could have a node for
each computer and an edge joining k
and m if there is a direct physical link
connecting them.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Information networks

Information networks. The World Wide
Web can be naturally viewed as a
directed graph, in which nodes
correspond to Web pages and there is
an edge from p to q if p has a hyperlink
to q.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Social networks.
Social networks. Given any collection of
people who interact for example friends,
we can define a network whose nodes
are people, with an edge joining two
nodes if they are friends.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Dependency networks.


Dependency networks. It is natural to define
directed
graphs
that
capture
the
interdependencies among a collection of
objects. For example, given the list of courses
offered by a college or university, we could
have a node for each course and an edge
from y to z if y is a pre-requisite for z.
These are only mere examples of graphs
there are plenty more to discuss but that will
not cover in this paper.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Applications
Applications of graph theory are primarily, but not exclusively,
concerned with labeled graphs and various specializations of
these. Many applications of graph theory exist in the form of
network analysis. These split broadly into two categories. First,
analysis to determine structural properties of a network, such as
the distribution of vertex degrees and the diameter of the graph.
A vast number of graph measures exist, and the production of
useful ones for various domains remains an active area of
research. Secondly, analysis to find a measurable quantity
within the network, for example, for a transportation network, the
level of vehicular flow within any portion of it. Graph theory
applications is also used in the studies of molecules in chemistry
and physics.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Circuits and Paths
Circuits and Paths
A Euler circuit in a graph G is a simple
circuit containing every edge of G. An
Euler path in G is a simple path
containing every edge of G
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Euler's Theorems
Theorem 1: If a graph has any vertices of odd degree, then it
CANNOT have an EULER CRCUIT and if a graph is connected
and every vertex has even degree, then it has AT LEAST ONE
EULER CIRCUIT.
Theorem 2: If a graph has more than 2 vertices of odd degree, then
it CANNOT have an EULER PATH and if a graph s connected
and has exactly 2 vertices of odd degree, then it has AT LEAST
ONE EULER PATH. Any such path must start at one of the odddegree vertices and end at the other.
Theorem 3: The sum of the degree of all the vertices of a graph is
an even number (exactly twce the number of edges). In every
graph, the number of vertices of odd degree must be even.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Figure 3
Number of ODD Vertices
Implication
graph)
(for
a
connected
0
There is at least one Euler Circuit
1
THIS IS IMPOSSIBLE
2
There is no Euler Circuit but at
least 1 Euler Path
More than 2
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
DNA Sequencing

Deoxyribonucleic acid, or DNA is a nucleic acid
molecule that contains the genetic instructions used
in the development and functioning of all living
organisms. The main role of DNA is the long-term
storage of information and it is often compared to a
set of blueprints, since DNA contains the instructions
needed to contruct other components of cells, such
as proteins and RNA molecules. The DNA segments
that carry this genetic information are called genes,
but other DNA sequences have structural purposes,
or are involved in regulating the use of this genetic
information.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
How is DNA Sequencing Done

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i)
chromosomes, which range in size from 50 million to 250
million bases, must first be broken into much shorter pieces.
ii)
Each short piece is used as a template to generate a set of
fragments that differ in length form each other by a single base.
iii)
The fragments in a set are separated by gel
electrophoresis. New fluorescent dyes allow separation of all
four fragments in a single lane on the gel.
iv)
The final base at the end of each fragment is identified.
This process recreates the original sequence of As, Ts, Cs, and
Gs for each short piece generated in the first step. Automated
sequencers analyze the resulting electropherograms, and the
output is a four-color chromatogram showing peaks that
represent each of the four DNA bases. After the bases are
“read”, computers are used to assemble the short sequences
into long continuous stretches that are analyzed for errors, genecoding regions, and other characteristics.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007


Euler Paths and DNA Sequencing
Euler helped changed the DNA world. With Euler’s
Paths, Circuits and Theorems, it changed the repeat
problem faced in DNA..
Even a single misassembly forces biologists to
conduct total genome screening for assembly errors.
Euler bypasses the “repeat problem,” because the
Eulerian Superpath approach transforms imperfect
repeats into different paths in the de Bruijn graph. As
a result, Euler does not even notice repeats unless
they are long perfect repeats.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Ecosystems

Ecosystems (ecological systems) are functional units
that result from the interactions of abiotic, biotic, and
cultural components. Like all systems they are a
combination of interacting, interrelated parts that form
a unitary whole. All ecosystems are “open” systems in
the sense that energy and matter are transferred in
an out. In this paper food webs will be used instead of
the ecosystem as a whole, to show how graph theory
is incorporated into this study.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Food Chain

A food web extends a food chain
concept from a simple linear pathway to
a complex network of interactions. The
best way to understand this concept is
through visualization. Below is a pitcure
of a food web.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Food Chain
In the graph Figure 4, this is an example of a Directed graph that pertains to
the ecosystem. As you can see this graph displays everyday natural animal
and insects consumption. For example the Grasshopper eats the Preying
Mantis. The arrows indicate in which direction the consumption takes place.
This is a common yet easy way to understand how the ecosystem and graph
theory are closely related.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Conclusion
So that in conclusion the Graph Theory Applications in Relation to the
Study of Ecosystems and DNA 2007 Team has arrived to the decision
that Euler’s Path was fundamental in DNA sequencing. Elulers Path
allowed for no repeats in DNA sequencing, which means that they were
not even identifiable in the sequence. Graph Theory is fundamental
when identifying possible correlations between mathematical modeling.
Graph theory can be compared to an If else statement in Computer
Science.
Graph Theory is essential when identifying highways and ecosystems
path. Graph Theory is also incorporated within our everyday life with
the Flow of Energy for example. The Graph Theory Applications in
Relation to the Study of Ecosystems and DNA 2007 team obtain our
goal of gaining an enhanced knowledge of Graph Theory, Euler path,
ecosystems and conducting useful and meaningful research.
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
References
• [1] KLI Theory Lab April 19, 2001, from the World Wide Web: http
http://www.kli.ac.at/theorylab/AuthPage/R/RosenR.html
• [2] IBM, Retrieved April 22, 2005, from the World Wide Web:
http://domino.research.ibm.com/comm/pr.nsf/pages/news.20000815_quantum.ht
ml
• [3] Ecosystems Educator Reference , Retrieved March 10, 2007, from the World
Wide Web: http://www.eduref.org/Virtual/Lessons/Science/Ecology/ECL0200.html
• [4] Mathematical Medicine and Biology, Retrieved April 5, 2007, from the World
Wide Web: http://imammb.oxfordjournals.org/cgi/content/abstract/6/1/1-a
• [5] Danel Sanders , Retrieved April 3, 2007, from the World Wide Web:
http://www1.cs.columbia.edu/~sanders/graphtheory/people/random.cgi?Sanders,
+Danel+P.
• [6] Euler's theorem - Wikipedia, the free encyclopedia, Retrieved April 20, 2005,
from the World Wide Web: http://en.wikipedia.org/wiki/Euler%27s_theorem
• [7] Graph Theory Research, Retrieved March 13, 2005, from the World Wide
Web: http://www1.cs.columbia.edu/~sanders/graphtheory/research/
• International Congress of Mathematicians Madrid 2006 Retrieved February 28,
2007, from the World Wide
Web:http://www.icm2006.org/scientificprogram/scientifisections/
Applications of Graph Theory and Ecosystems
Discrete Methods Group Project 2007
Questions…