1. MAT 299
Professor
Christopher French
and Karen Shuman
Steps Toward a Proof of Self-Duality of Hypercubes
Rachel Knak
Introduction
Definition (Graph)
A graph G consists of a vertex set V together with an edge set E. Two vertices v
and w on the graph G are adjacent if there exists an edge, vw ∈ E between them.
A graph G in which every pair of vertices is adjacent is a complete graph. A complete
graph with n vertices is denoted Kn. A Hamming graph is a graph H(n, r) such
that each vertex represents a sequence with r entries from {1, 2, . . . , n}, with two
vertices adjacent if they differ in one entry. The graph H(2, r) can be geometrically
represented by an n-dimensional cube. The graph H(n, 1) is a complete graph.
Definition (Distance Regular)
Let G be a graph with vertex set V . We say that a graph G is distance regular if for
each triple (i, j, k) there exists a constant ak
i,j such that for every pair of vertices v
and w with d(v, w) = k, there exist ak
i,j vertices u such that d(v, u) = i, and
d(w, u) = j. The constants ak
1,j are the structure constants of the graph G and can
be arranged in a matrix known as the structure matrix which takes the form of the
following (d + 1) × (d + 1) matrix.
a0
1,0 a1
1,0 · · · ad
1,0
a0
1,1 a1
1,1 · · · ad
1,1
a0
1,2 a1
1,2 · · · ad
1,2
...
... ... ...
a0
1,d a1
1,d · · · ad
1,d
All Hamming graphs are distance regular.
Example ( H(2, 3) ) Example ( K10) Example ( H(2, 2) )
Definition (Adjacency Matrix)
Let G be a graph with vertex set V containing n
vertices. The adjacency matrix of the graph G is
an n × n matrix which takes the form
a ,m = 1 − δ ,m =
0 if m ∈ E
1 if m ∈ E
.
Such that a ,m is the , mth
entry of A.
Example (Adjacency Matrix
of H(2, 2))



0 1 1 0
1 0 0 1
1 0 0 1
0 1 1 0




Definition (Idempotent)
We say a matrix A is an idempotent if A · A = A.
Let G be a graph with n vertices and let A be the adjacency matrix of G with
eigenvalues λ1, λ2, . . . , λg. An orthonormal basis for the eigenspace of λi can be used
to compute an idempotent, denoted Ei. The idempotent Ei represents a projection
onto the eigenspace of the eigenvalue i.
Definition (Krein Parameters)
Let A be the adjacency matrix of a graph G with eigenvalues λ1, λ2, . . . , λg with
corresponding idempotents E1, E2, . . . , Eg.Then the Krein parameters of the graph
G are the values bk
ij in the below equation
Eλh
∗ Eλj
=
1
n
g
k=0
bk
i,jEλk
where ∗ represents entry-wise multiplication and j, h ∈ N and j, h ≤ g.
Complete Graphs
When researching complete graphs we computed eigenvalues and eigenvectors for the
adjacency matrices of Kn, 1 ≤ n ≤ 8 using the computer program Mathematica and then
computed the idempotents by hand. We then identified trends in this data to construct
generalized formulas of eigenvectors, eigenvalues, and idempotents. After that, we used
other facts concerning complete graphs and symmetric matrices to prove these formulas
and, ultimately, to prove that all complete graphs are self dual (we say that a graph G is
self dual if its Krein parameters are equal to its structure constants).
This process is particularly well suited to complete graphs because each complete graph
has diameter 1, an adjacency matrix in which each entry is 1 save the main diagonal
on which each entry is 0. It follows that there are trends within the adjacency matrices
of complete graphs: A(Kn) has 2 eigenvalues and there are clear cut trends among
the eigenvectors. We then used this same strategy during our attempts to construct
generalized formulas for the eigenvectors, eigenvalues, and idempotents of the Hamming
graphs H(2, n). However, the research regarding the graphs H(n, 1) was considerably
more simple than the research regarding the Hamming graphs H(2, n) given the more
complex nature of these graphs.
Hypercube Characteristics
Here we will list a few of the results we proved during our investigations into the charac-
teristics of hypercubes.
Theorem
The adjacency matrix of the graph H(2, n) such that n ∈ N and n ≥ 3 takes the form
A(H(2, n − 1)) I2n
2
I2n
2
A(H(2, n − 1))
Theorem
Let H(2, n) be a Hamming graph such that n ∈ N. The structure matrix of H(2, n)
takes the form 



0 1 0 0 · · · 0 0 0
n 0 2 0 · · · 0 0 0
0 n − 1 0 3 · · · 0 0 0
0 0 n − 2 0 · · · 0 0 0
...
...
...
... ... ...
...
...
0 0 0 0 · · · 0 n − 1 0
0 0 0 0 · · · 2 0 n
0 0 0 0 · · · 0 1 0




Lemma
For a Hamming graph, H(2, n), n ∈ N, the eigenvalue n corresponds to the idempotent
En which takes the form 



1
2n
1
2n
1
2n · · · 1
2n
1
2n
1
2n
1
2n · · · 1
2n
... ... ... ... ...
1
2n
1
2n
1
2n · · · 1
2n



 .
Theorem
Let −→v be an eigenvector for the matrix A(H(2, n − 1)) such that
−(n − 1)−→v = A(H(2, n − 1))−→v and n ≥ 5, n ∈ N.
Let −→w =
−−→v
−→v
.
Then, −n−→w = A(H(2, n))−→w .
Conjectures
When attempting to determine whether the graphs H(2, n) were self dual, we
concluded that the graphs H(2, n), n ∈ N and n ≤ 5 are self dual. After concluding
this, we attempted to show that all graphs H(2, n) are self dual, but quickly realized
that this was no small venture. We then decided to deconstruct this problem into
smaller parts, looking to the eigenvalues, eigenvectors, and idempotents.
We created a few conjectures regarding the spectra of these matrices.
1 The eigenvalues follow the pattern n, n − 2, . . . , −n.
2 The multiplicities of the eigenvalues follow Pascal’s triangle.
3 Eigenvalues with equal magnitudes have equal multiplicities.
We also examined the idempotent En−2 closely.
The idempotent E2 for the graph H(2, 4) is as follows:
The graphic above is annotated to aid the viewer in seeing the patterns. The lines
connect entries of the same value.
The first thing that stood out to us on this matrix is that each entry is an eigenvalue
of A(H(2, 4)). As we examined this matrix further, we found that it could be broken
to 8 × 8 block matrices matrices, that those 8 × 8 block matrices could be broken
down into 4 × 4 block matrices, that the 4 × 4 matrices could be broken down into
2 × 2 block matrices, and finally that those 2 × 2 block matrices could be broken
down into 1 × 1 block matrices.
Upon further examination, we found that these block matrices had a pattern within
them, which led to the algorithm below.
Algorithm for Computing the Idempotent En−2 Corresponding to H(2, n)
(i) Start with the 1 × 1 matrix n .
(ii) Now use this matrix to compute a 2 × 2 matrix which takes the form
n n − 2
n − 2 n
(iii) Repeat the previous step n − 1 times so that the process has been completed a
total of n times, providing a 2n
× 2n
matrix. This 2n
× 2n
matrix is En−2.
Questions for Further Research
1 Is the aforementioned algorithm for the iteration of En−2 for n-dimensional
hypercubes true?
2 Are the conjectures regarding the eigenvalues of the adjacency matrices of the
hypercubes true?
3 What are the patterns for the all of the idempotents for the hypercubes?
4 Are all hypercubes self dual?
Acknowledgements
I would like to thank Sean Cates, Dan Davis, Lizzie Eason, Thomas Estabrook, and
Caleb Leedy