MC302 GRAPH THEORY Thursday, 9904

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MC302 GRAPH THEORYThursday, 9/9/04

• Today Opening ceremonies intro to course and
  graph theory
• Reading Chapter through Theorem 1.12, p. 7
• Exercises p. 15 ff. 1, 3, 7, 9
• First hand-in assignment will be given on Tuesday.

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Course Organization

• Class meetings Mixture of presentation of new
  material using slides and chalkboard group and
  individual work on problems
• Exercises
• not to hand in discuss at next class
• Includes assigned problems and leftover group
  work
• Homework About 4 assignments and some worksheet
  write-ups, 40 of grade
• Exams 2 in-class, 30 of grade
• Paper/talk 20 of grade
• Participation 10 of grade
• Participation (come to class) (arrive on
  time) (contribute ideas and questions)

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The AAA Approach to Graph Theory

• Applications
• Includes scheduling problems, network design,
  display of data, many others. Many applications
  in computer science
• Algorithms
• Many problems can be solved by modeling with
  graphs, then applying a graph algorithm
• Analysis
• The underlying theory of graphs, describing and
  predicting their properties
• Well take side trips to review proof
  techniques and counting techniques as needed.

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Some Introductory Examples

• Examples you sent me!
• US OPEN Starts with 128 players how many
  matches are there altogether?
• Given the data below,
• Whats the shortest route that starts in New
  York, visits all the other cities, and ends back
  in New York?

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Definition of a graph

• A graph consists of two sets, G (V, E)
• V is any nonempty finite set. The elements of V
  are called the vertices (or nodes) of G. V is
  called the vertex set of G.
• Intuitively, we think of the vertices as dots.
• E is any set (possible empty) of unordered pairs
  of vertices from V. Elements of E are called
  edges of G. E is called the edge set of G.
• Intuitively, we think of an edge as a line, or
  curve, connecting a pair of vertices.
• Text phrases this slightly differently.

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Some Important Graphs

• Kn -- the complete graph on n vertices
• Empty or Trivial graph -- no edges
• Pn -- the path on n vertices
• Cn -- the cycle on n vertices
• Sn -- the star on n vertices
• Petersen Graph pictured above
• Bipartite graph V(G) is partitioned into subsets
  X and Y such that all edges join a vertex of X
  with a vertex of Y
• Km, n -- the complete bipartite graph on n
  vertices

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Graph terminology

• Simple graph
• Adjacent vertices, neighbors
• Degree of a vertex
• Incident edges
• Isolated vertex
• Maximum degree ?(G) and minimum degree ?(G).
• Degree sequence

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Some results about degrees

• Theorem A (simple) graph G with n vertices has
  ?(G) n 1.
• Theorem In every graph there are at least two
  vertices with equal degree.
• Theorem (1.12) Let G be a graph with n vertices
  and m edges. Then