1. 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

2. 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.

3. 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.
Encryption is the conversion of data into a form, called a ciphertext that cannot be easily understood by unauthorized people. Decryption is the process of converting encrypted data back into its original form, so it can be understood.

4. Leonhard Euler (1707-1783) Leonhard Euler Theory
Notable Individual Leonhard Euler ( ) 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 He then returned to St. Petersburg, where he remained for the rest of his life.
In the evolution of cryptography, there are many noteworthy contributors. As early as BC, Julius Caesar used a simple substitution with the normal alphabet in government communications. In 1623, Sir Francis Bacon described a bilateral cipher, which now bears his name. Today, it is known as a 5-bit binary encoding. In the 1790’s, Thomas Jefferson, one of our founding fathers, was the originator of a wheel cipher. Today, it is known as a 5-bit binary encoding. Cipher is a small application to keep your secrets really secret by enciphering them with one of the most reliable encryption algorithms. William Frederick Friedman, founder of Riverbank Laboratories, cryptanalyst for the US government, and lead code-breaker of Japan’s World War II Purple Machine.

5. Examples of Graphs Figure 1.1 Figure 1.2 Simple Graph Directed Graph
Figure 1.1 Figure 1.2 Simple Graph Directed Graph
In the 1970’s, the precursor was established into the family of ciphers known as ‘Feistel ciphers’ named after Dr. Horst Feistel. The National Security Agency (NSA) worked with the Feistel ciphers to establish FIPS PUB-46, known today as Data Encryption Standard (DES). IBM submitted the proposed Data Encryption System (DES) at the invitation of the National Bureau of Standards, in an effort to develop secure electronic communication facilities for businesses such as banks and other large financial organizations

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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

13. 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 odd-degree 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.

14. Figure 3 Number of ODD Vertices Implication (for a connected graph)
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
The cryptology - many forms, from simple ciphers such as the Caesar Cipher, to complex, Computations performed in Elliptical Curve Cryptography.
The most widely used today involves the use of "Public" and "Private" keys. A key is a mathematical relation between two parties that allows for encryption and decryption. The private keys are held while the public keys are published for use by anyone. This is known as an asymmetric key system as a pair of different keys is being used. A symmetric system would involve the use of only one key passed from the sender to the receiver.
The technique most often used today for asymmetric keys is the Rivest-Shamir-Adleman (RSA) system. RSA is used today in Secure Sockets Layers, Firewalls, ATM Machines, and other secure systems. This system uses two large prime numbers to ensure that hackers cannot easily factor their product. This system is secure with a 512 bit key size, but many recommend 1024 bits or higher to ensure security.

15. 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.
The RSA system utilizes modular arithmetic working only with positive integers less than a chosen value. This chosen value is called the modulus and fixes the counting scheme to 0,1, 2 … (modulus -1), 0,1, 2, … (modulus-1).
Addition, subtraction, and multiplication can be performed using modular math, but division cannot. Modular math can be recognized by the expression: x = y(mod m). This expression is read: "x is equivalent to y, modulo m". The connotation of this expression is that when divided my m, x or y produces the same remainder. An example would be the expression 4 = 10 (mod 3). Both 4 and 10 have a remainder of 1 when divided by 3.
Another basic principle of modular mathematics is the expression: x + kp = x (mod p) with x being any positive integer, k being any constant, and p being a modulo prime number. For example, set x to 3 and p to 2 (3 +k(2) = 3 (mod 2)). It makes no difference k is set to; the solution will be the same.

16. How is DNA Sequencing Done
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, gene-coding regions, and other characteristics.
Starting with the fact that a prime number is any number that is divisible only by its self and 1, we can move onto the next step being coprimes.
Coprimes are comprised of a pair of numbers with the characteristic that their greatest common divisor is one and they are not equal (gcd (ni, nj) = 1 whenever i ≠ j). These numbers do not have to be prime as in the pair, 10 and 11, but distinct (unequal) prime numbers will always be coprime to each other.

17. 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.
Starting with the fact that a prime number is any number that is divisible only by its self and 1, we can move onto the next step being coprimes.
Coprimes are comprised of a pair of numbers with the characteristic that their greatest common divisor is one and they are not equal (gcd (ni, nj) = 1 whenever i ≠ j). These numbers do not have to be prime as in the pair, 10 and 11, but distinct (unequal) prime numbers will always be coprime to each other.

18. 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.
Starting with the fact that a prime number is any number that is divisible only by its self and 1, we can move onto the next step being coprimes.
Coprimes are comprised of a pair of numbers with the characteristic that their greatest common divisor is one and they are not equal (gcd (ni, nj) = 1 whenever i ≠ j). These numbers do not have to be prime as in the pair, 10 and 11, but distinct (unequal) prime numbers will always be coprime to each other.

19. 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.
Euler's theorem states that if n is a positive integer and a is coprime to n, then aφ(n) ≡ 1 (mod n) where φ(n) denotes Euler's totient function.
The totient, φ(n), of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. For example, φ(12) = 5 since the four numbers 1, 3, 5, 7 and 11 are coprime to 12.
In RSA, the theorem is used to reduce large powers of modulo n.

20. Food Chain
Euler's theorem states that if n is a positive integer and a is coprime to n, then aφ(n) ≡ 1 (mod n) where φ(n) denotes Euler's totient function.
The totient, φ(n), of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. For example, φ(12) = 5 since the four numbers 1, 3, 5, 7 and 11 are coprime to 12.
In RSA, the theorem is used to reduce large powers of modulo n.
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.

21. 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.
Euler's theorem states that if n is a positive integer and a is coprime to n, then aφ(n) ≡ 1 (mod n) where φ(n) denotes Euler's totient function.
The totient, φ(n), of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. For example, φ(12) = 5 since the four numbers 1, 3, 5, 7 and 11 are coprime to 12.
In RSA, the theorem is used to reduce large powers of modulo n.

22. References
[1] KLI Theory Lab April 19, 2001, from the World Wide Web: http
[2] IBM, Retrieved April 22, 2005, from the World Wide Web:
[3] Ecosystems Educator Reference , Retrieved March 10, 2007, from the World Wide Web:
[4]  Mathematical Medicine and Biology, Retrieved April 5, 2007, from the World Wide Web:
[5] Danel Sanders , Retrieved April 3, 2007, from the World Wide Web:
[6] Euler's theorem - Wikipedia, the free encyclopedia, Retrieved April 20, 2005, from the World Wide Web:
[7]  Graph Theory Research, Retrieved March 13, 2005, from the World Wide Web:
International Congress of Mathematicians Madrid 2006 Retrieved February 28, 2007, from the World Wide Web:

23. Questions…