Graphs

1
Chapter 9

• Graphs

2
Sec 9.1

• Graphs and Graph Models

3
Simple Graph

• A simple graph G is an ordered pair (V,E),
  consisting of a nonempty set of vertices V, and a
  set of edges E, where each edge e?E corresponding
  to a unique element in u,v u?V, v?V, u ? v
  
• For an edge e which corresponds to u,v the
  vertices u and v are called the endpoints of this
  edge

4
Multigraph

• A multigraph G is an ordered pair (V,E),
  consisting of a nonempty set of vertices V, and a
  set of edges E, where each edge e ? E corresponds
  to some element in u,v u?V, v?V, u ? v
• Note It is possible for two edges to correspond
  to the same u,v , allowing for two or more
  edges to have the same endpoints.

5
Pseudograph

• A pseudograph G is an ordered pair (V,E),
  consisting of a set of vertices V, and a set of
  edges E, where each edge e ? E corresponds to
  some element in u,v u?V, v?V ? u,u
  u?V
• Note that an edge may correspond to u thus
  allowing both endpoints to be the same. Such
  edges are called loops.

6
Directed Graph

• A directed graph G is an ordered pair (V,E),
  consisting of a set of vertices V, and a set of
  directed edges E, where each edge e ? E
  corresponds to an element in (u,v) u?V, v?V
• Note Associating directed edges to ordered
  pairs means that each edge has an associated
  orientation or direction. If e corresponds to
  (u,v), then u is called the initial vertex and v
  is called the terminal vertex of e.

7
Simple Directed Graph

• A directed graph G is simple if it has no loops
  and no multiple directed edges. So a simple
  directed graph is an ordered pair (V,E),
  consisting of a set of vertices V, and a set of
  edges E, where each edge e ? E corresponds to at
  a unique element in (u,v) u?V, v?V, u ? v

8
Directed Multigraph

• A directed multigraph G is an ordered pair (V,E),
  consisting of a set of vertices V, and a set of
  edges E, where each edge e ? E corresponds to
  some element in (u,v) u?V, v?V
• Note More than one edge may correspond to the
  same (u,v) allowing multiple directed edges
  between two vertices.

9
Summary Graph G (V,E)

• Simple Each edge ei ? a unique u,v, u ? v
• Multigraph Each edge ei ? some u,v, u ? v
• Pseudograph Each edge ei ? some u,v or u
• Simple Directed Graph Each edge ei ? a unique
  (u,v), u ? v
• Directed Multigraph Each edge ei ? some (u,v)

10
Homework

• Sec 9.1
• pg. 596 1, 3, 5, 7, 9, 15, 21

11
Sec 9.2

• Graph Terminology

12
Definitions

• Adjacent Vertices Two vertices, u and v in an
  undirected graph G are adjacent (or neighbors) if
  there is an edge e in E which corresponds to
  u,v
• Incident If edge e corresponds to u,v, the
  edge e is called incident with the vertices u and
  v.
• Connected If edge e corresponds to u,v, the
  vertices u and v are said to be connected.
• Endpoints If edge e corresponds to u,v, the
  vertices u and v are said to be endpoints of e

13
Degree of a Vertex

• Deg(v) The degree of a vertex in an undirected
  graph G is the number of edges incident with it.
  (with the convention that a loop at v contributes
  twice to the degree of that vertex)
• The Handshaking Theorem. Let G (V,E) be an
  undirected graph. The sum of the degrees of all
  the vertices of the graph is equal to 2E.
• Theorem An undirected graph has an even number
  of vertices of odd degree.

14
Definition Directed Graphs

• Let G is a directed graph with an edge e
  corresponding to (u,v),
• Adjacent Then vertex u is said to be adjacent to
  v and vertex v is said to be adjacent from u.
• Initial and Terminal Vertices Vertex u is called
  the initial vertex and vertex v is called the
  terminal vertex of edge e.
• For a loop in a directed graph, the single vertex
  is considered both the initial and the terminal
  vertex.

15
In-degree and Out-degree

• Deg-(v) In a directed graph G, the in-degree of
  vertex v is the number of edges with v as their
  terminal vertex.
• Deg(v) In a directed graph G, the out-degree
  of vertex v is the number of edges with v as
  their initial vertex.
• Theorem For any directed graph G (V,E), If
  the summation is taken over all v ? V, ?deg(v)
  ?deg-(v) E2e, e of edges

16
Some Special Simple Graphs

• Complete A simple graph G is complete if it
  contains an edge between each pair of vertices.
  A complete graph of n vertices is denoted by Kn.
• Bipartite A simple graph G (V,E) is called
  bipartite if the vertices in V can be partitioned
  into two disjoint sets V1 and V2 so that each
  edge e ? E has exactly one endpoint in V1 and the
  other in V2.

17
Results

• Theorem A simple graph is bipartite if and only
  if it is possible to assign one of two different
  colors to each vertex of the graph so that no two
  adjacent vertices are assigned the same color.
• Complete Bipartite A bipartite graph whose
  vertices can partitioned into two sets of m and n
  vertices respectively with an edge between every
  pair of vertices in the two sets is called a
  complete bipartite graph, denoted by Km,n.

18
Application Job Assignments

• Suppose that there are m employees in a group and
  j different jobs that need to be done where m ?
  j. Each employee is trained to do one or more of
  there j jobs.
• Vertices
• Employees
• Jobs
• Edges connect employees to jobs

19
Homework

• Sec 9.2
• pg. 608 1, 3, 5, 7, 9, 17, 21, 23