Essential Question:
How do we use matrices to solve problems?
•How do we relate matrices to vertex-edge graphs?
Standards:
MM3A7: Students will understand and apply matrix
representations of vertex-edge graphs.
•MM3A7a: Use graphs to represent
realistic situations.
•MM3A7b: Use matrices to
represent graphs, and solve
problems that can be
represented by graphs.
Math 3 Activating Strategy
Vertex-Edge Graph
Adjacency Matrix
A B
A
D
A
B
B
C
E
C
D
F
G
E
F
G
C
D
E
F G
Math 3 Activating Strategy
Vertex-Edge Graph
A
D
B
C
E
A B
C
A
0
1
1
B
1
0
1
C
D
F
G
Adjacency Matrix
E
F
G
D
E
F G
AN OKEFENOKEE FOOD WEB Learning Task
Recent weather conditions have caused a dramatic
increase in the insect population of the Okefenokee
Swamp area. The insects are annoying to people and
animals and health officials are concerned there will
be an increase in disease. Local authorities want to
use an insecticide that would literally wipe out the
entire insect population of the area.
You, as an employee of the Environmental Protection
Agency (EPA), must determine how detrimental this
would be to the environment. Specifically, you are
concerned about the effects on the food web of six
animals known to populate the swamp.
Consider the following directed
vertex-edge graph (digraph) of a food web for the six animals and
the insects that are causing the problem.
Here each vertex represents an animal or insects. The direction of the edges indicates
whether an animal preys on the linked animal. For example, raccoons eat fish. (Note: The
food web shown is simplified. Initial producers of nutrients, plants, have not been
included.)
Adjacency matrices can be used in conjunction with
digraphs. If we consider just the relationships between
raccoons, fish, and frogs in the food web shown, an
adjacency matrix would be
R
R
FT
F
FT F
0 1 1 
0 0 0 


0 1 1
1. Construct the associated matrix F to represent this web.
M
B
I
M
C
B
B
R
M
F
FT
I
C
FT
F
R
M
I
C
FT
F
R
1. Construct the associated matrix F to represent this web.
What does a row containing a single one indicate?
What does a column of zeros indicate?
2. Which animals have the most direct sources of food?
How can this be determined from the matrix?
Find the number of direct food sources for each animal.
3. The insect column has the most ones. What does this
suggest about the food web?
More Practice with Vertex-Edge Graphs Worksheet
To save paper, an office administrator is trying to deliver messages through oral
communication between coworkers. The administrator tells one employee the
message; the administrator then considers the message to be deliverable to every
worker who has direct contact (talking directly to the designated employee) or
second-hand contact (talking to someone who had direct contact with the
designated employee).
The vertex-edge graph below summarizes the direct contacts made between the six
employees supervised by the administrator. Complete the adjacency matrix below to
summarize the direct contacts made between the six employees.
Ann
Ann Bob Caf Dan Eve Fin
Ann
Bob
Caf
Bob
Caf
Dan
Eve
Fin
Dan
Eve
Fin
Complete the table below to show which employees will and will not get the
message if the administrator tells the employee in the first column.
Administrator tells:
Who gets the message?
Who doesn’t get the
message?
Ann
Bob
Caf
Dan
Eve
Fin
Ann
Ann Bob Caf Dan Eve Fin
Ann
Bob
Caf
Bob
Caf
Dan
Eve
Fin
Dan
Eve
Fin