applications of graph theory and ecosystems discrete methods group project 2007 application of graph...

Applications of Graph Theory and Ecosystems Discrete Methods Group Project 2007

Application of Graph Theory and Ecosystems

Discrete Methods Group Project 2007 Discrete Methods Group Project 2007 Erika Mizelle, Kaiem L. Frink,Erika Mizelle, Kaiem L. Frink,

Elizabeth City State UniversityElizabeth City State University

1704 Weeksville Road1704 Weeksville Road

Elizabeth City, North Carolina 27909Elizabeth City, North Carolina 27909


Abstract

The word graph (graf) comes from the Greek The word graph (graf) comes from the Greek word graphein and is a noun. It is a diagram word graphein and is a noun. It is a diagram indicating any sort of relationship between indicating any sort of relationship between two or more things by means of a system of two or more things by means of a system of dots, curves, bars, or lines. The word dots, curves, bars, or lines. The word ecosystem (e’ko sis’tem) is from the Greek ecosystem (e’ko sis’tem) is from the Greek word oikos meaning habitat + system. It is word oikos meaning habitat + system. It is defined as a community of organisms and defined as a community of organisms and their nonliving environment. their nonliving environment.


Introduction of Graph Theory

Applications of graph theory are primarily, but Applications of graph theory are primarily, but not exclusively, concerned with labeled not exclusively, concerned with labeled graphs and various specializations of these. graphs and various specializations of these. Graphs can be used in almost any field of Graphs can be used in almost any field of study for various different reasons. This paper study for various different reasons. This paper will discuss how graph theory and its will discuss how graph theory and its applications can be used in ecosystems and applications can be used in ecosystems and DNA sequencing.DNA sequencing.


Leonhard Euler (1707-1783) Leonhard Euler Theory Notable IndividualNotable Individual

Leonhard Euler (1707-1783) Leonhard Euler was the son of a Leonhard Euler (1707-1783) Leonhard Euler was the son of a Calvinist minister from the vicinity of Basel, Switzerland. At 13 Calvinist minister from the vicinity of Basel, Switzerland. At 13 he entered the University of Basel, pursing a career in theology, he entered the University of Basel, pursing a career in theology, as his father wished. At the University of Basel, Johann as his father wished. At the University of Basel, Johann Bernoulli of the famous Bernoulli family of mathematicians Bernoulli of the famous Bernoulli family of mathematicians tutored Euler. His interest and skills led him to abandon his tutored Euler. His interest and skills led him to abandon his theological studies and take up mathematics. Euler obtained his theological studies and take up mathematics. Euler obtained his masters degree in philosophy at the age of 16. In 1727 Peter masters degree in philosophy at the age of 16. In 1727 Peter the Great invited him to join the Academy at St. Petersburg. In the Great invited him to join the Academy at St. Petersburg. In 1736, Euler solved a problem known as the Seven Bridges of 1736, Euler solved a problem known as the Seven Bridges of Konigsberg. In 1741 he moved to the Berlin Academy, where he Konigsberg. In 1741 he moved to the Berlin Academy, where he stayed until 1766. He then returned to St. Petersburg, where he stayed until 1766. He then returned to St. Petersburg, where he remained for the rest of his life. remained for the rest of his life.


Examples of Graphs

Figure 1.1Figure 1.1Figure 1.2Figure 1.2

Simple Graph Directed GraphSimple Graph Directed Graph


Transportation networks

Transportation networks.Transportation networks. The map of a The map of a bus line route forms a graph. The nodes bus line route forms a graph. The nodes (vertices) could represent the different (vertices) could represent the different cities or states that the bus visits.cities or states that the bus visits.


Communication network.

Communication network.Communication network. A collection of A collection of computers that are connected via a computers that are connected via a communication network can be naturally communication network can be naturally modeled as a graph in a few different modeled as a graph in a few different ways. First, we could have a node for ways. First, we could have a node for each computer and an edge joining k each computer and an edge joining k and m if there is a direct physical link and m if there is a direct physical link connecting them.connecting them.


Information networks

Information networks.Information networks. The World Wide The World Wide Web can be naturally viewed as a Web can be naturally viewed as a directed graph, in which nodes directed graph, in which nodes correspond to Web pages and there is correspond to Web pages and there is an edge from p to q if p has a hyperlink an edge from p to q if p has a hyperlink to q.to q.


Social networks.

Social networks. Given any collection of Social networks. Given any collection of people who interact for example friends, people who interact for example friends, we can define a network whose nodes we can define a network whose nodes are people, with an edge joining two are people, with an edge joining two nodes if they are friends. nodes if they are friends.


Dependency networks.

Dependency networks.Dependency networks. It is natural to define It is natural to define directed graphs that capture the directed graphs that capture the interdependencies among a collection of interdependencies among a collection of objects. For example, given the list of courses objects. For example, given the list of courses offered by a college or university, we could offered by a college or university, we could have a node for each course and an edge have a node for each course and an edge from y to z if y is a pre-requisite for z.from y to z if y is a pre-requisite for z.

These are only mere examples of graphs These are only mere examples of graphs there are plenty more to discuss but that will there are plenty more to discuss but that will not cover in this paper.not cover in this paper.


Applications

Applications of graph theory are primarily, but not exclusively, Applications of graph theory are primarily, but not exclusively, concerned with labeled graphs and various specializations of concerned with labeled graphs and various specializations of these. Many applications of graph theory exist in the form of these. Many applications of graph theory exist in the form of network analysis. These split broadly into two categories. First, network analysis. These split broadly into two categories. First, analysis to determine structural properties of a network, such as analysis to determine structural properties of a network, such as the distribution of vertex degrees and the diameter of the graph. the distribution of vertex degrees and the diameter of the graph. A vast number of graph measures exist, and the production of A vast number of graph measures exist, and the production of useful ones for various domains remains an active area of useful ones for various domains remains an active area of research. Secondly, analysis to find a measurable quantity research. Secondly, analysis to find a measurable quantity within the network, for example, for a transportation network, the within the network, for example, for a transportation network, the level of vehicular flow within any portion of it. Graph theory level of vehicular flow within any portion of it. Graph theory applications is also used in the studies of molecules in applications is also used in the studies of molecules in chemistry and physics. chemistry and physics.


Circuits and Paths

Circuits and PathsCircuits and Paths

A Euler circuit in a graph G is a simple A Euler circuit in a graph G is a simple circuit containing every edge of G. An circuit containing every edge of G. An Euler path in G is a simple path Euler path in G is a simple path containing every edge of Gcontaining every edge of G


Euler's Theorems

Theorem 1Theorem 1: If a graph has any vertices of odd degree, then it : If a graph has any vertices of odd degree, then it CANNOT have an EULER CRCUIT and if a graph is connected CANNOT have an EULER CRCUIT and if a graph is connected and every vertex has even degree, then it has AT LEAST ONE and every vertex has even degree, then it has AT LEAST ONE EULER CIRCUIT.EULER CIRCUIT.

Theorem 2Theorem 2: If a graph has more than 2 vertices of odd degree, then : If a graph has more than 2 vertices of odd degree, then

it CANNOT have an EULER PATH and if a graph s connected 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 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 odd-ONE EULER PATH. Any such path must start at one of the odd-degree vertices and end at the other.degree vertices and end at the other.

Theorem 3Theorem 3: The sum of the degree of all the vertices of a graph is : The sum of the degree of all the vertices of a graph is

an even number (exactly twce the number of edges). In every an even number (exactly twce the number of edges). In every graph, the number of vertices of odd degree must be even. graph, the number of vertices of odd degree must be even.


Figure 3

Number of ODD VerticesNumber of ODD Vertices Implication (for a connected Implication (for a connected graph)graph)

00 There is at least one Euler CircuitThere is at least one Euler Circuit

11 THIS IS IMPOSSIBLETHIS IS IMPOSSIBLE

22 There is no Euler Circuit but at There is no Euler Circuit but at least 1 Euler Pathleast 1 Euler Path

More than 2More than 2


DNA Sequencing

Deoxyribonucleic acid, or DNA is a nucleic acid Deoxyribonucleic acid, or DNA is a nucleic acid molecule that contains the genetic instructions used in molecule that contains the genetic instructions used in the development and functioning of all living organisms. the development and functioning of all living organisms. The main role of DNA is the long-term storage of The main role of DNA is the long-term storage of information and it is often compared to a set of information and it is often compared to a set of blueprints, since DNA contains the instructions needed blueprints, since DNA contains the instructions needed to contruct other components of cells, such as proteins to contruct other components of cells, such as proteins and RNA molecules. The DNA segments that carry this and RNA molecules. The DNA segments that carry this genetic information are called genes, but other DNA genetic information are called genes, but other DNA sequences have structural purposes, or are involved in sequences have structural purposes, or are involved in regulating the use of this genetic information.regulating the use of this genetic information.


i)i) chromosomes, which range in size from 50 million to 250 chromosomes, which range in size from 50 million to 250

million bases, must first be broken into much shorter pieces.million bases, must first be broken into much shorter pieces. ii)ii) Each short piece is used as a template to generate a set of 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.fragments that differ in length form each other by a single base. iii)iii) The fragments in a set are separated by gel The fragments in a set are separated by gel

electrophoresis. New fluorescent dyes allow separation of all electrophoresis. New fluorescent dyes allow separation of all four fragments in a single lane on the gel.four fragments in a single lane on the gel.

iv)iv) The final base at the end of each fragment is identified. The final base at the end of each fragment is identified. This process recreates the original sequence of As, Ts, Cs, and This process recreates the original sequence of As, Ts, Cs, and Gs for each short piece generated in the first step. Automated Gs for each short piece generated in the first step. Automated sequencers analyze the resulting electropherograms, and the sequencers analyze the resulting electropherograms, and the output is a four-color chromatogram showing peaks that output is a four-color chromatogram showing peaks that represent each of the four DNA bases. After the bases are represent each of the four DNA bases. After the bases are “read”, computers are used to assemble the short sequences “read”, computers are used to assemble the short sequences into long continuous stretches that are analyzed for errors, gene-into long continuous stretches that are analyzed for errors, gene-coding regions, and other characteristics.coding regions, and other characteristics.

How is DNA Sequencing DoneHow is DNA Sequencing Done


Euler helped changed the DNA world. With Euler’s Euler helped changed the DNA world. With Euler’s Paths, Circuits and Theorems, it changed the repeat Paths, Circuits and Theorems, it changed the repeat problem faced in DNA.. problem faced in DNA..

Even a single misassembly forces biologists to Even a single misassembly forces biologists to conduct total genome screening for assembly errors. conduct total genome screening for assembly errors. Euler bypasses the “repeat problem,” because the Euler bypasses the “repeat problem,” because the Eulerian Superpath approach transforms imperfect Eulerian Superpath approach transforms imperfect repeats into different paths in the de Bruijn graph. As repeats into different paths in the de Bruijn graph. As a result, Euler does not even notice repeats unless a result, Euler does not even notice repeats unless they are long perfect repeats. they are long perfect repeats.

Euler Paths and DNA SequencingEuler Paths and DNA Sequencing


Ecosystems (ecological systems) are functional units Ecosystems (ecological systems) are functional units

that result from the interactions of abiotic, biotic, and that result from the interactions of abiotic, biotic, and cultural components. Like all systems they are a cultural components. Like all systems they are a combination of interacting, interrelated parts that form combination of interacting, interrelated parts that form a unitary whole. All ecosystems are “open” systems in a unitary whole. All ecosystems are “open” systems in the sense that energy and matter are transferred in the sense that energy and matter are transferred in an out. In this paper food webs will be used instead of an out. In this paper food webs will be used instead of the ecosystem as a whole, to show how graph theory the ecosystem as a whole, to show how graph theory is incorporated into this study.is incorporated into this study.

EcosystemsEcosystems


Food Chain

A food web extends a food chain A food web extends a food chain concept from a simple linear pathway to concept from a simple linear pathway to a complex network of interactions. The a complex network of interactions. The best way to understand this concept is best way to understand this concept is through visualization. Below is a pitcure through visualization. Below is a pitcure of a food web.of a food web.


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.


ConclusionSo 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.


References• [1][1] KLI Theory Lab April 19, 2001, from the World Wide Web: http KLI Theory Lab April 19, 2001, from the World Wide Web: http

http://www.kli.ac.at/theorylab/AuthPage/R/RosenR.htmlhttp://www.kli.ac.at/theorylab/AuthPage/R/RosenR.html• [2][2] IBM, Retrieved April 22, 2005, from the World Wide Web: IBM, Retrieved April 22, 2005, from the World Wide Web:

http://domino.research.ibm.com/comm/pr.nsf/pages/news.20000815_quantum.hthttp://domino.research.ibm.com/comm/pr.nsf/pages/news.20000815_quantum.htmlml

• [3][3] Ecosystems Educator Reference , Retrieved March 10, 2007, from the World Ecosystems Educator Reference , Retrieved March 10, 2007, from the World Wide Web: http://www.eduref.org/Virtual/Lessons/Science/Ecology/ECL0200.htmlWide Web: http://www.eduref.org/Virtual/Lessons/Science/Ecology/ECL0200.html

• [4][4] Mathematical Medicine and Biology, Retrieved April 5, 2007, from the World Mathematical Medicine and Biology, Retrieved April 5, 2007, from the World Wide Web: http://imammb.oxfordjournals.org/cgi/content/abstract/6/1/1-aWide Web: http://imammb.oxfordjournals.org/cgi/content/abstract/6/1/1-a

• [5][5] Danel Sanders , Retrieved April 3, 2007, from the World Wide Web: Danel Sanders , Retrieved April 3, 2007, from the World Wide Web: http://www1.cs.columbia.edu/~sanders/graphtheory/people/random.cgi?Sanders,http://www1.cs.columbia.edu/~sanders/graphtheory/people/random.cgi?Sanders,+Danel+P.+Danel+P.

• [6][6] Euler's theorem - Wikipedia, the free encyclopedia, Retrieved April 20, 2005, Euler's theorem - Wikipedia, the free encyclopedia, Retrieved April 20, 2005, from the World Wide Web: http://en.wikipedia.org/wiki/Euler%27s_theoremfrom the World Wide Web: http://en.wikipedia.org/wiki/Euler%27s_theorem

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• International Congress of Mathematicians Madrid 2006 Retrieved February 28, International Congress of Mathematicians Madrid 2006 Retrieved February 28, 2007, from the World Wide 2007, from the World Wide Web:http://www.icm2006.org/scientificprogram/scientifisections/ Web:http://www.icm2006.org/scientificprogram/scientifisections/


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