GRAPH THEORY

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GRAPH THEORY

• Heather Shelton
• December 6, 2005

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What is Graph Theory?

• Introduced in the 18th Century by Leonhard Euler
• Konisberg Bridge Problem
• Graph is a collection of dots that may or may not
  be connected by lines

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Terms to Know

• Edge Any line that is drawn from one dot to
  another dot
• Vertex each dot that appears in the
  collaboration of dots
• degree of the vertex of the graph is the number
  of edges that touch it
• size of the graph is how ever many vertices the
  graph has

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More Terms

• A graph is said to be regular if ever vertex has
  the same degree
• A path is a route that is traveled along edges
  and through vertices in a graph
• All of the vertices and edges in a path are
  connected to one another
• A cycle is a path which begins and ends on the
  same vertex
• A cycle is sometimes called a circuit
• A node is a connecting point at which several
  lines come together

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Graphs

• Graphs are structures that contain vertices and
  edges that are connected from each vertex
• Different types of graphs are determined by the
  number of edges that are connected by vertices
• Types of Graphs
• Simple Graph
• Multigraph
• Pseudograph
• Directed Graph

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Simple Graph

• A simple graph can be thought of as G (V,E)
  where V is a non empty set of vertices and E is a
  set of unordered pairs of distinct elements of V
  called edges
• Example network of computers and telephone lines
  between computers

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Multigraph

• G (V,E) consists of a set V of vertices and a
  set E of edges, and a function f from E to
  u,v u,v is an element of V, and u cannot
  equal v. The edges e1 and e2 are then called
  multiple or parallel edges

Not every multigraph can be a simple graph but
every simple graph can be a multigraph.
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Pseudograph

• Psuedographs are the most general types of
  undirected graphs
• These graphs also have a loop in them
• Also considered to be a branch of the simple graph

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Directed Graphs

• Directed graphs and directed multigraphs are
  similar in that each vertex is an ordered pair
• multiple edges in the same direction are not
  allowed but loops are

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Graph Models

• Niche Overlap Graphs
• Most often found when dealing with animals
• i.e. Food Web
• Influence Graph
• Identifies power and influence structure
• i.e. Umbrella Example
• Round Robin Model
• Most well known by people interested in sports
• i.e. Athletic Tournament

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Influential and Round RobinModels
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Special Simple Graphs

• Called complete graphs because each contains one
  edge between each pair of distinct vertices
• Other type of special simple graphs contain
  cycles- basic geometric figures
• Wheels occur when a point is put into the middle
  of the figure and edges from each of the outer
  vertices join at the new point in the middle

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Complete Graphs
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Cycles-Geometric Figures
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Wheels
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Traveling Salesperson

• Use graph theory to find the quickest way to
  complete the entire route without retracing any
  steps
• A team of mathematicians discovered that there
  are 653,837,184,000 different routes
• After 92 hours they were finally able to come up
  with the shortest route without retracing steps

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Relation to Graph Theory

• When a salesperson can get from every city to
  every other city directly then the graph is said
  to be complete
• When a salesperson travels to every city and his
  last stop is the spot where he started his
  expedition, the trip is said to be a round-trip
• The length of a tour or of travel is the sum of
  the lengths of the lines in a round trip

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Graph Coloring

• Started with assigning a color to each vertex
• Then noted that number of areas could be colored
  using fewer colors than number of vertices
• The goal here was to not color each adjoining
  area with the same color

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So what does this mean?

• Graph theory is a big part of everyones every
  day life
• Most people have been doing graph theory since
  childhood
• i.e. coloring a picture, geometric shapes,
  figuring out driving distances for a trip
• Everywhere around us are graphs and graph theory