Vocabulary and Representations of Graphs

1
Vocabulary and Representations of Graphs
2
NC Standard Course of Study

• Competency Goal 1 The learner will use matrices
  and graphs to model relationships and solve
  problems.
• Objective 1.02 Use graph theory to model
  relationships and solve problems.

3
Graphs

• Recall that a graph is a set of points called
  vertices and a set of line segments called edges.
• Often graphs are used to model situations in
  which the vertices represent objects, and edges
  are drawn between the vertices on the basis of a
  particular relationship between the objects.
• The important characteristics of a graph will
  remain unchanged if the edges are curved.

4
Explore This

• Suppose the following diagram represents the
  starting five players on a high school basketball
  team, and the edges denote friendships.

B
A
E
C
D
5
Exploration (contd)

• This graph indicates that player C is friends
  with all of the other players and that E had only
  two friends, C and B.
• Note that edge CE and edge DB intersect in this
  graph but that their intersection does not create
  a new vertex.

6
Graph Questions

1. Which player has only one friend?
2. How many friends does E have? Who are they?
3. Redraw the graph so that A has no friends.

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Exploration (contd)

• Consider the following solutions
• A.
• Two, C and B.
• The graph shown below.

B
A
E
C
D
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Connected Graphs

• The previous graph is not connected.
• A graph is connected if there is a path between
  each pair of vertices.
• In the previous graph, there was no path from
  point A to any of the other vertices.

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Explore This

• Lets say that the graph instead represents rooms
  in the school.
• The vertices are connected if there are direct
  hallways between two rooms.
• According to the graph, a student can get from
  room C directly to any of the other four rooms.

B
A
E
C
D
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Adjacent Vertices

• When two vertices are connected with an edge they
  are said to be adjacent.
• C is adjacent to A, B, D, and E.
• Although there is no direct route from D to room
  A, it is possible to get from room D to room A by
  going through room C.
• Although a path exists between D and A, they are
  not adjacent.

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Redrawing the Graph

• Lets try redrawing the graph so there is direct
  access from each room to every other room.

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Possible Solutions
A
B
A
E
C
D
D
B
C
E
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Similarities

• Even though the graphs appear to be different,
  they are structurally the same, so they are
  considered to be the same graph.

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Complete Graphs

• Graphs in which every pair of vertices is
  adjacent, are called complete graphs.
• Complete graphs are often denoted by KN, where N
  is the number of vertices in the graph.
• The previous graphs are deciptions of a K5 graph.

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Other Ways to Represent Graphs

• There are other ways to represent graphs besides
  a diagram.
• A second method is to list the set of vertices
  and the set of edges. This can be illustrated
  as
• Vertices A, B, C, D, E
• Edges AC, CB, CE, CD, BD, BE

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Adjacency Matrix

• This is the third type of way to represent a
  graph.
• It is used to represent the vertices and edges of
  the graph in a computer.
• A 5 X 5 matrix is formed by labeling the rows and
  columns corresponding to the vertices. If an
  edge exists between vertices, a 1 will appear in
  the position in the matrix otherwise a 0 will
  appear.

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Adjacency Matrix (contd)
A
B
C
D
E
A
B
C
D
E
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Adjacency Matrix (contd)

• The entry in row 2, column 4 is a 1, which
  indicates that an edge exists between vertices B
  and D.

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Policy Change

• From now on, for all practice problems, they need
  to be written down and completed to be turned in
  the day after we finish a section.
• Please do them on paper which can be turned in
  (so preferably, not on NOTES!)

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Practice Problems

1. Mr. Butler bought six different types of fish.
   Some of the fish can live in the same aquarium,
   but others cannot. Guppies can live with
   Mollies Swordtails can live with Guppies Gold
   Rams can live only with Plecostomi and Piranhas
   cannot live with any of the other fish. Draw a
   graph to illustrate this.

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Practice Problems (contd)

• Construct a graph for each of the following sets
  of vertices and edges. Which of the graphs are
  connected? Which are complete?
• a. VA, B, C, D, E b. VM, N, O, P, Q, R,
  S
• EAB, AC, AD, AE, BE E MN, SR, QS,
  SP, OP
• c. VE, F, G, J, K, M d. VW, X, Y, Z
• EEF, KM, FG, JM, EG, KJ EWX, XZ, YZ, XY, WZ,
  WY

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Practice Problems (contd)

• Draw a diagram representing the graph with
  vertices A, B, C, D, E, F and edged AB, CD,
  DE, EC, EF.
• a. Name two vertices that are not adjacent.
• b. F, E, C is one possible path from F to C.
  This path has length of 2, since two edges were
  traveled to get from F to C. Name a path from F
  to C with a length of 3.
• c. Is this graph connected? Why or why not?
• d. Is this graph complete? Why or why not?

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Practice Problems (contd)

• Draw a graph with 5 vertices in which vertex W is
  adjacent to Y X is adjacent to Z and V is
  adjacent to each of the other vertices.
• Construct a graph for each adjacency matrix.
  Label the Vertices A, B, C, .

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Practice Problems (contd)

• a.
• b.

• c.

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Practice Problems (contd)

• Determine an adjacency matrix for each of the
  following graphs

A
B
P
O
R
D
C
M
S
N
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Practice Problems (contd)

• Give the adjacency matrix for the following
  graph

W
Y
V
Z
X
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Practice Problems (contd)

1. What do you notice about the main diagonal of the
   matrix?
2. A Matrix may be symmetric with respect to one of
   its rows, columns or diagonals. Does the matrix
   above possess symmetry? If so, where?
3. What would a 1 on the main diagonal indicate?
   What would a 2 in the second row, first column,
   indicate?

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Practice Problems (contd)

• 8. Using the graph and the adjacency matrix in
  exercise 7, find the sum of each row of the
  matrix. What does the sum of the rows tell you
  about the graph?
• 9. The number of edges that have a specific
  vertex as an endpoint is know as the degree or
  valence of that vertex. In the graph on the next
  slide, the degree of vertex W, denoted by deg(W)
  is 4. Find the degree of each of the other
  vertices.

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Practice Problems (contd)
W
Y
V
X
Z
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Practice Problems (contd)

• An edge that connects a vertex to itself is
  called a loop. If a graph contains a loop or
  multiple edges (more than one edge between two
  vertices), the graph is know as a multigraph.
• a. Give the adjacency matrix for the following
  multigraph

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Practice Problems (contd)
A
D
C
E
B
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Practice Problems (contd)

• What is the degree of each of the five vertices?
• Complete the chart below for the sum of the
  degrees of the vertices in a complete graph.

Graph of Vertices Sum of Degrees of all of the Vertices Recurrence Relation
K1 1 0
K2 2 2 T2T1 2
K3 3 6 T3T24
K4
K5
K6
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Practice Problems (contd)

• Write a recurrence relation that expresses the
  relationship between the sum of the degrees of
  all of the vertices for KN and the sum for KN-1.