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. Abstract.
Discrete Methods Group Project 2007
Erika Mizelle, Kaiem L. Frink,
Elizabeth City State University
1704 Weeksville Road
Elizabeth City, North Carolina 27909
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 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.
Notable IndividualLeonhard 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.
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.
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.
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 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.
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
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.
Euler helped changed the DNA world. With Euler’s Paths, Circuits and Theorems, it changed the repeat problem faced in DNA..
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.
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.