San Francisco State University
Alyssa Palfreyman
Adjacencies in Three-Dimensional Graphs:
Matrices and Marshmallows
Introduction
This lesson was designed for the San Francisco Math Circle as the conclusion to a series on graph theory.
Three-dimensional polytopes are introduced to the students as three-dimensional graphs (a collection of
vertices and edges). The students investigate the structure and purpose of an adjacency matrix in relation to
its graph, and eventually “build” their own three-dimensional graphs given an adjacency matrix. This lesson
was also given to a math analysis class over two 48-minute periods. As a two-day lesson, the students had
opportunity for greater investigation of the extension questions and richer discussions.
Learning Objectives
Students will
• Learn another representation of mathematical relationships using matrices
• Be able to represent the adjacencies of a graph by writing its adjacency matrix
• Be able to construct a graph given an adjacency matrix
• Understand that objects can be represented algebraically/abstractly
Common Core standards for mathematical practice being used in this lesson:
1. Make sense of problems and perservere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
Background
This lesson was designed for students who had previously worked with graphs and were familiar with the
definition of adjacent vertices. However, this background knowledge is not essential to success in this lesson.
Adequate time can be spent at the beginning to define these terms and familiarize students with these concepts.
A graph G is a collection of vertices and edges where each edge connects exactly two vertices. Examples of
three graphs are shown below.
•
•
•
•
•
•
•
•
•
•
1
•
•
•
Adjacent vertices are two vertices that are connected by an edge. A labeled graph is a graph in which the
vertices are labeled 1, 2, . . . as in the example below.
•4
2•
•3
•1
Most students will not know formally what a matrix is. This object is introduced in the lesson as an object for
organizing numbers in a tabular form. Each entry is described by its row and column; the entry in the ith row
and jth column is the (i, j) entry. Knowing matrix operations is not required.
In the first part of this lesson, students are building an understanding of and providing a definition for an
adjacency matrix of a graph. The formal definition of an adjacency matrix AG of a graph G is as follows. Let
ai j be the (i, j) entry of AG . Then,
(
1 if vertex i is adjacent to vertex j
ai j =
0 otherwise
(Note: a vertex is not adjacent to itself.) The adjacency matrix for the labeled graph shown above is
0
1
0 1 1 0
B1 0 1 1C
C
AG = B
@1 1 0 1A .
0 1 1 0
For a more detailed introduction to graph theory, I direct you to [3]. For more specific information on
algebraic graph theory such as adjacent matrices, see [2].
Materials Required
For each group of four:
• Set of 4 “three-dimensional graphs,” made out of zometools [1] (cube, octahedron, triangular pyramid,
triangular prism)
2
• Set of 4 adjacency matrices each on a separate slip of paper (see Adjacency Matrices of 3-Dimensional
Graphs attached)
00 1 0 1 1 0 0 01
00 1 1 1 1 01
00 1 1 1 0 0 1
1
0
1
0
0
1
0
0
✓0 1 1 1◆
1
0
1
0
1
1
0
1
0
1
0
0
1
0C
1
0
1
0
1
0
B
1
0
1
0
0
0
0
1C
1
1
0
0
0
1A
@
a : 11 01 10 11
b : @11 10 01 10 01 11A g : B
d
:
1
0
0
0
1
1
@10 01 00 00 01 10 01 10A
1
1
1
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
1
1
1
0
0
0
0
0
1
0
0
1
0
1
1
0
0
1
1
0
0
0
1
1
1
0
• Dot stickers (for students to label the vertices on the zometool graphs) - optional
• Marshmallows (about 8-12 per group)
• Straws with the ends cut sharp (about 20 per group)
• Cleaning wipes (to wipe the tables clean after the activity)
For each student:
• Matrices and Marshmallows Worksheet (see attached)
Instructional Plan
• (3 min) Introduce the students to a matrix. Draw a sample matrix on the board, such as
0
1
1 3
4 2
@2
1 0 1A ,
1 2
7 3
and describe the matrix in terms of rows and columns. For example the shown matrix has three rows
and four columns. Each entry in the matrix can be expressed by the row and column that it is in. For
example, the entry 0 in the shown matrix is the entry in the 2nd row and 3rd column. We can call this
the (2, 3) entry. In general, the entry in the i-th row and j-th column is the (i, j) entry.
• (3 min) Remind students of the concept of a graph. Draw a sample graph of the board emphasizing the
vertices and the edges, such as the following graph with 4 vertices and 5 edges.
•4
2•
•3
•1
Explain that it is often useful to label the vertices to make it easier to talk about the graph. Label the
sample graph with any choice of labeling (e.g. 1, 2, 3, . . . , or a, b, c, . . . ). Ask for volunteers to give an
example of two vertices on the sample graph that are adjacent (such as 1 $ 2 or 2 $ 3). Similarly,
ask for examples of vertices that are not adjacent (such as 1 6$ 4). Next, introduce the students to the
three-dimensional graphs they will be working with in this activity. Show them one of the polyhedra
made out of zomes, emphasizing that it has vertices and edges just like when they worked with graphs
two-dimensionally.
3
• (20-30 min) Hand out the group sets of graphs made of zometools and the sets of adjacency matrices.
The group task is to match each matrix with one of the graphs, and explain as a group why they think
they have the correct matchings. At this point, the students don’t know that these specific matrices
are called adjacency matrices, and they don’t know how they relate to the graphs. The students will
make conjectures about how the matrices match with the graphs. If they have trouble getting started,
ask them to make observations about the matrices and graphs: Number of rows/columns? How many
vertices/edges/faces in this graph? How many 1’s in a particular row? They might say things like “this
matrix matches with this graph because they are both the biggest.” Encourage them to describe more
clearly what they mean by the “biggest”: the most vertices? the most edges? the most faces? If they
think the number of rows/columns in the matrix matches with the number of faces, point out that the
triangular pyramid has 5 faces and none of the matrices have 5 rows/columns. If students continue to
struggle, give hints like labeling the rows and columns for them (e.g. 1-6), or labeling the vertices of a
graph (with the dot stickers).
When a group has a general idea of how the matrices match with the graphs, give them the handout and
ask them to complete the front page.
• (5 min) When most of the groups have figured out the connection from the graphs to the matrices, have
a class discussion on their findings.
• (20 min) Pass out the marshmallows and straws so they can build their own 3-dim graphs. The second
page of the handout has four new adjacency matrices from which they can build their graphs. They
can write with markers on the marshmallows if they like so that their vertices are labeled. Have them
complete the second page of the handout. There are extension questions provided for the students that
work faster than others.
4
Reflection
I enjoyed teaching this lesson very much. I was initially worried that students would get frustrated or
check-out during the first group activity when they match up the matrices and graphs. But each time I taught
this lesson, all the students were engaged and genuinely interested in finding the relationship between the
graphs and matrices. It was a good opportunity for students to make conjectures and try to justify their
guesses. At times, the students would make an incorrect claim and the whole group would be convinced that
it was correct because it might work for one case. These times actually provided good opportunites for rich
discussion about finding counterexamples and proving/testing conjectures.
When students are building the graphs out of marshmallows and straws, they had to work together to be able to
both hold it together physically, as well as keep track of their work. I liked this encouragement of team work. I
observed one group of four very bright students that preferred to work independently. This made it hard to get
them to collaborate and discuss ideas with each other during the first activity. However, as soon as they had to
build the graphs as a team, they quickly realized they could not do it alone and they worked together quite well.
I found that even with the faster-paced classes, there is little to no time left for the extension questions given
a one hour time frame. The students were able to build 2-3 of the graphs out of marshmallows on the second
page of the handout. A more in-depth discussion and greater investigation of the extension questions could
take place with more time. If given the time and flexibility, I think this lesson works better when split into
two days or extended over a longer class period.
Last note: The marshmallows do get quite sticky after a while so be prepared with with paper towels or some
kind of cleaner.
References
[1] Zometool. http://zometool.com/.
[2] Norman Biggs. Algebraic graph theory. Cambridge Mathematical Library. Cambridge University Press,
Cambridge, second edition, 1993.
[3] Douglas B. West. Introduction to graph theory. Prentice Hall Inc., Upper Saddle River, NJ, 1996.
5
Adjacency Matrices of 3-Dimensional Graphs
There are two sets of the adjacency matrices below. Make enough copies for each group to have one set. Cut
each matrix into separate slips and give one set to each group.
(For the instructor’s reference: a = triangular pyramid, b = octahedron, g = cube, d = triangular prism)
0
0
B1
B
@1
1
0
0
B1
B
B1
B
B1
B
@1
0
0
0
B1
B
B0
B
B1
B
B1
B
B0
B
@0
0
1
0
1
0
0
1
0
0
0
0
B1
B
B1
B
B1
B
@0
0
1
0
1
0
1
1
0
1
0
1
0
0
1
0
1
0
1
0
1
0
1
0
1
1
1
1
0
1
0
1
1
0
1
0
0
0
0
1
1
1
0
0
0
1
1
1
0
1
1
0
1
0
1
1
1
0
0
0
0
1
0
1
1
0
0
0
1
1
1
1
1C
C
1A
0
1
1
0
1
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
1
0
0
B1
B
@1
1
a
1
0
1C
C
1C
C
1C
C
1A
0
0
B1
B
B1
B
B1
B
@1
0
b
0
0
0
1
0
0
1
0
1
1
0
0C
C
0C
C
1C
C
1C
C
0C
C
1A
g
0
1
0
0C
C
1C
C
1C
C
1A
0
d
1
0
1
1
1
0
1
0
1
1
1
1
1C
C
1A
0
1
1
0
1
1
1
0
1
0
1
1
0
1
0
1
1
1
1
0
1
0
1
a
1
0
1C
C
1C
C
1C
C
1A
b
0
0
0
B1
B
B0
B
B1
B
B1
B
B0
B
@0
0
1
0
1
0
0
1
0
0
0
1
0
1
0
0
1
0
1
0
1
0
0
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
1
0
1
0
0
0
B1
B
B1
B
B1
B
@0
0
1
0
1
0
1
0
1
1
0
0
0
1
1
0
0
0
1
1
0
1
0
1
0
1
1
0
0C
C
1C
C
1C
C
1A
0
0
0
1
0
0
1
0
1
1
0
0C
C
0C
C
1C
C
1C
C
0C
C
1A
g
0
d
Matrices and Marshmallows
Your group has been given four matrices that each match up with a (3-dimensional) graph. Once
your group has agreed which matrix matches with each graph, draw a picture of the graph below
the matrix that it matches (or describe in words what it looks like).
0
0
B
B1
B
B1
@
1
1
0
1
1
a
1
1
0
1
1
1
C
1C
C
1C
A
0
0
0
B
B1
B
B1
B
B
B1
B
B1
@
0
1
0
1
0
1
1
1
1
0
1
0
1
b
1
0
1
0
1
1
1
1
0
1
0
1
0
0
B
B1
B
B0
B
B
B1
B
B1
B
B
B0
B
B0
@
0
1
0
C
1C
C
1C
C
C
1C
C
1C
A
0
1
0
1
0
0
1
0
0
0
1
0
1
0
0
1
0
1
0
1
0
0
0
0
1
g
1
0
0
0
0
1
0
1
0
1
0
0
1
0
1
0
0
0
1
0
0
1
0
1
1
0
C
0C
C
0C
C
C
1C
C
1C
C
C
0C
C
1C
A
0
0
0
B
B1
B
B1
B
B
B1
B
B0
@
0
1
0
1
0
1
0
1
1
0
0
0
1
d
1
0
0
0
1
1
0
1
0
1
0
1
• Describe in a few sentences how you know when a matrix matches with a graph.
We call each of these matrices the Adjacency matrix of its graph because it encodes when two
vertices are adjacent in the graph.
• How many rows are in the adjacency matrix that corresponds to the graph below? How many
columns?
• Write down the adjacency matrix.
1
1
0
C
0C
C
1C
C
C
1C
C
1C
A
0
Now using the marshmallows and straws provided, try to construct the graphs for each of the
following adjacency matrices. Once the graph is constructed, try to draw the graph in the box (or
describe in words what it looks like).
0
0
B
B1
B
B0
B
B
B0
B
B1
@
1
1
0
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
0
1
1
0
B
B1
B
B1
B
B
B1
B
B0
B
B
@0
0
l
1
0
0
1
0
1
1
1
0
0
1
0
1
1
C
1C
C
1C
C
C
1C
C
1C
A
0
1
0
0
0
1
1
0
0
1
0
1
0
0
1
0
0
1
1
0
0
1
x
0
0
B
B1
B
B1
B
B
@1
0
1
0
0C
C
C
1C
C
C
0C
C
1C
C
C
1A
0
0
0
B
B0
B
B1
B
B
B1
B
B1
@
0
1
0
1
1
1
1
1
1
0
1
0
1C
C
C
1C
C
C
1A
0
1
0
0
0
1
1
1
1
0
0
0
1
1
1
1
0
0
0
µ
0
0
0
1
1
1
1
1
1
0
1
1
s
1
0
C
1C
C
1C
C
C
1C
C
0C
A
0
Extension
1. All of the adjacency matrices you saw today were symmetric. That is, if you fold the matrix
along the diagonal, the entries will match up. Will an adjacency matrix always be symmetric?
Why or why not?
2. The graph that matched with d at the beginning is the same graph that matched with s above.
But d and s are different matrices. Explain how this is possible.
3. Another important matrix in graph theory is the Laplacian matrix. If two labeled vertices in
a graph are adjacent you put a -1 in the matrix, if two vertices are not adjacent you put a 0,
and along the diagonal you put the degree of the corresponding vertex (the number of edges
that touch that vertex). An example is shown below. Write the Laplacian matrices for the
graphs that matched with a, b , g, and d .
0
•5
1•
B
B
B
@
•4
2•
•3
Labeled Graph
2
2
1
0
0
1
1
3
1
0
1
0
1
3
1
1
Laplacian
0
0
1
2
1
1
1
1C
C
1C
1A
4