Matrix Analysis of Networks
(Application of Graph Theory)
Grade Level: 10 and 11
Teacher: Agbonma (Oby) Egbuonu
Objectives:
 Students will be able to draw and interpret finite graphs
 Students will be able to draw and interpret directed graphs
 Students will be able to plan routes for city streets and develop mathematical
reasoning
TEKS: b1B, b2A
b1B: In solving problems, the student collects data and records results, organizes the
data, makes scatter plots, fits the curves to the appropriate parent function, interprets
the results, and proceeds to model, predict, and make decisions and critical
judgments.
b2A: The student uses tools including matrices, factoring, and properties of
exponents to simplify expressions and transform and solve problems.
Vocabulary:
Define, draw or give example of:
 Network
 Finite graph
 Directed graph
 Vertex (node)
 Edge(s)
 Loop
Materials:




graph paper
pencil and paper
graphing calculator
ruler
Warm-up exercise:
Sample problem
The Speedy Delivery Company has the delivery area represented by the network shown
below:
6
4
C
B
5
A
7
3
5
3
D
F
5
4
7
E
4
G
The numerical weightings are the average driving times in minutes between two pick-up
locations. For example, on average it should take a driver 4 minutes to drive from
location A to location B.
1. On average how long should it take the driver to travel from A to B to C (path
ABC)?
Introduction:
Cooperative Learning
Work with a partner to plan and design a town
(a) Make a list of the four most important places or buildings for the town
(b) On a sheet of graph paper, draw a simple map of the town. Mark each building
and include roads to get from place to place.
(c) Make a table that indicates which pairs of locations you can travel between
without going through other locations.
Monitor students as they work in groups. This activity comes from Advanced
Algebra, Prentice Hall.
Discuss with students the following definitions:
Network
Finite graph
Vertices (nodes)
Directed graph
Network
A network consists of a set of points, called nodes (vertices), which are connected by
segments, called edges (arcs).
Edge
Node
In practical applications, the nodes often represent geographical points like cities,
intersections, railroad stops, pipeline connections or individual locations. The edges often
represent links between nodes, for example, roads between cities. The arcs can be
undirected (two ways) or directed (one way or two ways). A basic problem involving
networks is to find the shortest path between two given nodes.
Source : HSOR.org:modules-Speedy Delivery
Have students brainstorm for a list of networks.
Ask students to think about ways in which different kinds of networks are similar and
different.
Finite Graph
Finite graph is a set of points connected by paths. The points are called vertices. A matrix
can be used to describe a finite graph.
N1 (-6, 2); N2 (-1, 4); N3 (-2,-2) and N4 (4, 2)
5
-1, 4
4
3
-6, 2
4, 2
2
1
0
-8
-6
-4
-2
-1 0
2
4
-2, -2
-3
Using the finite graph above, show students how to present the information from the
finite graph in a matrix. Tell students to use a 1 only where there is a direct path
between two nodes, and use a 0 to indicate no path between two nodes (vertices).
Questions (Finite Graph)
(1) What do the zeros along the main diagonal of the matrix represent?
(2) What does the zero in position a41 indicate? Explain.
(3) State the lengths of N1N2, and N2N3.
6
Directed Graph
Directed graph is a finite graph that indicates the direction of a path.
You can use a matrix to write directed graph.
British Airways route:
Houston
London
New York
Using the British Airlines route map, write a matrix A that indicates the routes between
these cities.
Student practice
Exercises:
1. Crystal and Enrique each live near a train station. The train stops near both
houses. Vanessa drives a car and has permission to drive to Crystal and Enrique’s
houses. Enrique can go by bus to Vanessa’s house but Crystal cannot get to
Vanessa’s house by bus. Draw a directed graph indicating the travel options.
Then write a matrix to represent the information.
2. Scientists A, B, and C communicate by fax if the have each other’s fax numbers.
They communicate by electronic mail if they have each other’s e-mail addresses.
Matrix E models the scientists’ possible e-mail matrix, and matrix F models their
possible fax matrix.
1 1 0
1 1 1 = E


0 1 1
0 0 0 
1 0 1 = F


0 1 0
a) Draw directed graphs for each matrix.
b) Analyze the values along the main diagonals of E and F.
c) Calculate E + F. What does the matrix model? Explain.
3. Alice and Becky live on the River Drive East, at the intersections of Memorial
Bridge and Benjamin Bridge, respectively. Carl and David live on River Drive
West, at the intersections of Benjamin Bridge and Memorial Bridge, respectively.
River Drive East is a one-way street running east. River Drive West is one-way
running west. Both bridges are two-way. Draw a directed graph indicating road
travel between the houses. Then write a matrix to represent the information.
Assessment:
 Class participation
 Completed assignments
 Teamwork
Closure:
Writing: Compare and contrast a finite graph and a directed graph. Remind students that
compare means how they are alike, and contrast means how they are different.
Resources:
Advanced Algebra, Prentice Hall
Discovering Geometry, An Investigative Approach, Key Curriculum Press
HSOR.org: Modules (High School Operation Research)