Spectra of Planar Graphs and their Duals
Elizabeth Draves
June – July 2015
Abstract
Graph theory encompasses the study of graphs composed of vertices and edges,
without considering the relative location or distance between the vertices, but rather
focusing solely on connections. This paper will delve into graph theory by examining
planar graphs, or graphs without intersecting edges, and their duals. The study looks into
the function and relation of eigenvalues for these two types of graphs, specifically
regarding the magnitude of the eigenvalues for adjacency and Laplace matrices, and in
turn offers conjectures about these graphs.
Introduction
Graph theory is an actively explored field of mathematics which in modern day has
proven a beneficial tool in fields such as biochemistry and computer science. Spectral
graph theory which this paper will examine focuses more specifically on the study of
matrices used to code information about graphs including adjacency and Laplace matrices
and their associated eigenvalues. A graph simply consists of points, or vertices, which are
sometimes connected by line segments, or edges. Those vertices which are connected by
an edge are called adjacent, while the rest are considered non-adjacent. For example, the
graph below, K4, has four vertices, all with three connections, and thus all vertices are
adjacent to the others.
For all graphs, the adjacency matrices will always be a square matrix composed of
zeroes and ones. If a graph has n vertices, the adjacency matrix will contain n columns and
n rows. An adjacency matrix’s elements are either a zero or one, a zero representing a lack
of an edge between two vertices and a one demonstrating a connection, or adjacency,
between the two. In the graphs being examined, the adjacency matrix will always have
zeroes in the diagonal. For example, in K4, none of the vertices are connected to
themselves; hence zeroes in the diagonal. But all other vertices are connected, so there ones
are inserted in all other positions. Further, it is important to note that matrices will be
symmetric with respect to their diagonals. In a graph, a connection or lack thereof between
two vertices, k and n, will be represented twice in the adjacency matrix, once in the matrix’s
element (k,n) and also in the element (n,k).
Furthermore, it has been observed that for adjacency matrices, the summation of
all total eigenvalues of the matrix equals zero. Additionally, isomorphic graphs, or graphs
with the same number of vertices connected in the same fashion, will not be examined
individually because it is known that although these graphs may not have identical
adjacency matrices, they will still have the same set of eigenvalues.
Eigenvalues are values from which eigenvectors can be found. A vector is an
eigenvector of a square matrix A if A times vector v is a scalar multiple of v, or Av = λv for
some constant λ. In calculating eigenvalues, and in turn eigenvectors, the formula det(AλI)=0, or the detriment of the adjacency matrix minus lambda times the identity matrix,
can be used. The identity matrix is a matrix with all zeroes except in the diagonal, which
has ones. In this examination, eigenvalues of the adjacency matrices corresponding to
planar graphs will be found. To find the eigenvalues of graph 1, all the diagonal zeroes in
the matrix would be substituted with negative lambdas (to achieve A- λI), and then the
detriment of this new matrix would be found and set to zero.
det
=0
In taking the detriment, a polynomial with the variable lambda is created, and is set equal
to zero. From this characteristic polynomial the roots, or eigenvalues for the matrix, are
found. Alternatively, and in data collection for this research, the matrix was plugged into
the program Mathematica, to find the eigenvalues -1, -1, -1, and 3 are found.
There are other matrices associated with graphs. The Laplace matrix, which is
calculated from its degree matrix (D) minus its adjacency matrix (A), or D-A can also be
interesting. A degree matrix is a matrix with all zeros except in the diagonal, in which the
degree, or the vertex’s number of connections, is recorded. So for graph 1, the degree
matrix would be displayed as below because all vertices have three connections.
Planar graphs are those which can be drawn in such a way that none of their edges
cross. For any planar graph, Euler’s formula holds: V – E + F = 2, where V is the number
of vertices, E is the number of edges, and F is the number of faces. Graphs containing the
complete graph K5 or the complete bipartite graph K3,3 will never be planar graphs.
Graph theory is still being explored today because of its applications in our modern
world. Load sharing, which attempts to distribute workload across multiple computing
resources effectively, and similarly equitable resource distribution both rely on spectral
graph theory. In applying linear algebra to spectral graph theory, progress can be made in
understanding more about graphs and their characteristics.
This examination looks at the eigenvalues for planar graphs as well as the
eigenvalues for planar graph duals. A multigraph dual of graph G is a graph having one
vertex corresponding to each face of G, and an edge joining two neighboring faces between
each edge of G. Conclusions will be made as to characteristics of the eigenvalues of planar
graphs and their duals, and how these two types of graphs are related.
Research
In order to examine eigenvalues of planar graphs and their duals, nineteen planar
graphs were constructed, and respective adjacency and Laplace matrices were created.
Using the program Mathematica, eigenvalues were found from the matrices, which were
assembled in a table (see Appendix 1). From these values, patterns were observed, from
which theorems were stipulated and proven.
Observations
In examining planar graphs and their duals, the eigenvalues were collected and
compared. While there was not always an apparent pattern between eigenvalues of graphs
and their duals, tree graphs and cycle graphs specifically showed a prominent pattern. Tree
graphs are acyclic, undirected, simple connected graphs, and cycle graphs, notated as Cn,
are graphs which have a single cycle consisting of vertices in a closed chain. Looking at
adjacency matrix eigenvalues, it was noted in all example cases of tree and cycle graphs
that the sum of the squared eigenvalues was equal to the sum of the absolute values of the
eigenvalues of their duals. Or, more eloquently put, if the original graph is called C and its
dual is notated as C*, then:
We will prove this equality by showing that both terms in this equation are equivalent to
twice the number of edges of the original graph.
Theorems
Theorem 1: Given a matrix, A, the general characteristic polynomial found from
the equation det(A- λI)=0 is equal to (-1)n[λn- (λn-1)(tr(A))+ … +det(A)].
Proof:
We assume the equation det(A- λI)=0 where λs are the eigenvalues of the matrix A, and
know that the trace is equal to the sum of the diagonal of a matrix.
Creating a general 2x2 matrix, we get:
The detriment of this matrix is ad-bc, and its trace equals a+d. In taking A-λI, we then
have:
Then, taking its detriment and setting it equal to zero, thus following det(A- λI)=0, we get
the polynomial
(a-λ)(d-λ)-bc=0. Simplifying this gives us
λ2 –λ(a+d)+(ad-bc)=0, in which the lambda with the second highest exponent, or λ1, is
multiplied by the matrix’s trace, and the constant is the matrix’s detriment.
Using a generic 3x3 matrix, we get:
The trace of this matrix is a+e+i, and its detriment is (aei-ahf)-(bdi-bgf)-(cdh-cge).
Applying A- λI to the matrix it becomes:
We then want to take the detriment of this matrix and set it equal to zero, which is
Det(A- λI)=(a-λ)[(e-λ)(i-λ)-hf]-b[d(i-λ)-gf]+c[dh-g(e-λ)], which simplifies to
Det(A-λI)=- λ3+λ2(a+e+i)- λ(e+i+a+ei-hf+bd-cg)+ (aei-ahf-bdi+bgf-cdh+cge).
Noting the sign of the values, the first term, or the term with the highest exponent, is
negative, and then the sign switches for each consecutive term. Additionally, the second
highest exponent in this polynomial, in this case λ2, is multiplied by the trace of the
original matrix. Further, the constant is equal to the original matrix’s detriment. This
holds true in both the general 2x2 and 3x3 matrix.
After examining these two general matrices, the theorem can be applied to all other sized
matrices through induction, as shown in the textbook [citation here].
Thus, we can say that det(A-λI)= (-1)n[λn+(λn-1)(tr(A))+ … +det(A)].
The theorem above has been shown to be true.
Q.E.D.
Theorem 2: The trace of an adjacency matrix A is equal to the sum of the
eigenvalues of A.
Proof:
We know that to find a graph’s eigenvalues, we must set up the equation (A-λI)=0. From
theorem 1, this is known to be equal to (-1)n[λn- (λn-1)(tr(A))+ … +det(A)].
This equation, (-1)n[λn- (λn-1)(tr(A))+ … +det(A)], is a polynomial which when factored
must be equal to
(λ-λ1)( λ-λ2)…( λ-λn), where the λxs are the eigenvalues of the matrix.
In multiplying out this rewritten form of the polynomial, (λ-λ1)( λ-λ2)(…)( λ-λn), the
lambda with the highest exponent, λn, will have a constant of one in front of it, because it
will not be multiplied by any of the eigenvalues, but instead only by other lambdas.
However, the value including the lambda with the second highest exponent, λn-1, will have
a constant. This λn-1 term will be equal to:
(λ1 λn-1 + λ2 λn-1 + λ3 λn-1 + … + λn λn-1).
Or, rewritten
(λ1 + λ2 + λ3 + … + λn)(λn-1), or
Σλ (or sum of eigenvalues) times (λn-1).
Based off of theorem one, we know that in a matrix’s characteristic polynomial, the λn-1
term is multiplied by the trace of A.
Thus, this constant multiplied by λn-1 is equal to both Σλ and the tr(A). Therefore we can
conclude that
Σλ=tr(A).
Q.E.D.
Corollary 1: For any adjacency matrix A, we have tr(A2)= Σλ2.
Proof: From the theorem that Σλ=tr(A), it immediately follows that tr(A2)= Σλ2, as
shown below.
λv=Av by definition of an eigenvalue.
Λ2v=Aλv
λ2v=A2v.
Q.E.D.
Theorem 3: Given an adjacency matrix A for a graph G, each diagonal entry of the
matrix A2 is the degree of the corresponding vertex of G.
Proof:
A is a symmetric matrix. Each entry in A is either a one or a zero, representing an edge or
an absence thereof. We are looking at graphs that are not multigraphs, and therefore there
cannot be more than one connection between two vertices of the graph.
Creating a generic 3x3 matrix, we attain:
And in squaring this matrix, the three diagonal values are respectively a2+bd+ch,
bd+e2+if, and ch+fi+j2.
In each of these values, there is a squared term (i.e. a2, e2, and j2) which is equal to zero.
There are no loops in the graphs, and thus the adjacency matrices’ diagonals will always
have zeroes. Hence the squared term in each diagonal position will not contribute to its
magnitude.
Also, because the graphs for which the adjacency matrices are created are not
multigraphs, there will always be either a zero or one in every other position.
In a2+bd+ch, b and d are symmetric across the diagonal and thus are the same number
(either 0 or 1), as are c and h.
Since 02=0 and 12=1, the values for bd and ch are equal to the values of b and c
themselves. The first row contains the values of a, b, and c. a is equal to zero, and b and c
are accounted for in the terms bd and ch, which are added together in the matrix A2 to
attain a diagonal entry which is the sum of the values in the matrix’s first row. This sum
is equivalent to the degree of a vertex from the graph.
In examining bd+e2+if and ch+fi+j2, which are the other two terms in the squared
matrix’s diagonal, the same idea applies where the squared constant equals zero and the
other terms in the row, multiplied by their symmetric equivalent, are added together.
More generally, in the matrix A2, we can consider an element in the diagonal of the kth
row and kth column, ak,k. This element would be the sum of each element of the kth row of
the matrix A (ak1, ak2, etc.) multiplied by each element of the kth column of matrix A (a1k,
a2k, etc.). Since the matrix A is symmetric, the corresponding k values in the row and
column (for example ak1 and a1k) would be equivalent, being either a one or a zero. So,
ak,k of A2 would equal ak1a1k + ak2a2k + … + aknank. When these values are added up, the
total equals the number of connections for a vertex of A.
Thus, a squared matrix has diagonal values which are equivalent to the degrees of the
original graph’s vertices.
The theorem has therefore been proven.
Q.E.D.
Corollary 2: Tr(A2) = 2E.
Proof: It immediately follows from theorem 3 that when we add up the trace of
the matrix A2, we will attain the sum of the degrees of the original graph, and
Sum of degrees = 2E, E being edges in the graph by the handshake lemma, which states
that
.
This is true because the sum of the degree counts every edge exactly twice since each
edge is connected to two vertices.
From these theorems, we can see that Σλ2, where λs are the eigenvalues of an adjacency
matrix of a planar graph, is equivalent to the tr(A2), which in turn is equal to 2E, where E
is the number of edges of a graph.
Q.E.D.
We now need to examine the term ΣIλI, where λs are in this case equal to the eigenvalues
of an adjacency matrix of a multi-dual of a graph.
Theorem 4: Let G be a tree or a cycle with dual G*. The sum of the absolute
values of the eigenvalues of G*, is equal to twice the number of edges of the original
graph G.
Proof:
In examining duals of trees, there will only be one vertex in the dual because the original
graph only has one face. Thus, a 1x1 adjacency matrix will result.
For example, the dual of the tree graph below will only have one vertex, because no
interior faces are created by the graph.
Tree graph
Dual
The one element in the adjacency matrix of the dual, i.e. the number of connections
which the one vertex has to itself, will be equivalent to the one eigenvalue of the matrix.
This number of loops in a dual equals the number of edges in the graph, by definition. In
a tree graph’s dual, we consider one loop to be two edges for the purpose of creating the
adjacency matrix, because looking locally at the vertex a loop creates two separate edges.
So, the dual of the example tree will have one vertex with four loops coming off of it,
corresponding to each edge of the original graph. Thus, the adjacency matrix will equal
2(number of loops), or [8], with the eigenvalue of 8, which is twice the number of edges
of the original graph.
Therefore, the one eigenvalue of the tree’s dual is equal to two times the number of edges
of the original graph, which can be rewritten as
ΣIλI = 2E.
Cycle graphs also create easily described duals. The original cycle graphs have two faces
(one interior and one exterior), and each vertex has two connections.
The graph below demonstrates this principle.
Cycle Graph
Dual
Constructing the dual of a cycle graph, the two vertices will be connected with a line
through each edge of the original graph. In the example above, the dual would consist of
two vertices with five edges connecting them.
Since there are two vertices in the a cycle graph’s dual, the duals will always have two by
two adjacency matrices, with zeroes in the diagonal, as there are no loops, and the
number of edges of the original graph in the other two elements of the matrix. So for the
example above, the adjacency matrix of the dual would be:
In order to find the eigenvalues of this matrix, the detriment of (A-Iλ) is set equal to zero,
which becomes λ2 – (# edges)2 = 0. When the roots of this equation are taken, the
eigenvalues of this two by two matrix are
+(# edges of original graph), and –(# edges of original graph).
For our example, in taking the dual’s detriment, we would have the equation
λ2 – 25 = 0,
Which would make the eigenvalues +5 and -5, and ΣIλI would then equal 10,
Because in taking the ΣIλI, the number of edges of the original graph is multiplied by
two, thus being equal to 2E.
Q.E.D.
Note: In exploring tree graphs and their duals, we noted that tree graphs with the same
number of vertices did not have identical eigenvalues. However the Σλ2 for each set of
eigenvalues was identical, and thus the Σλ2 for the tree graph’s eigenvalues all were still
equal to ΣIλI of the dual. All trees with the same number of vertices had the same dual,
which was just one vertex with the same number of loops as there were edges in the original
graph.
Conclusion
In summation, because both Σλ2 of the adjacency matrix eigenvalues and ΣIλI of the
eigenvalues of the dual’s adjacency matrix are equivalent to two times the number of edges
of a graph, they are equal to each other.
Therefore, we can say that it was found and proven that for a graph’s adjacency matrix (C)
and its dual (C*),
.
During the project we observed other phenomenon we have yet to prove. For example, in
the tested graphs and their duals, it appeared that the Σλ2 of the adjacency matrix of a graph
is always equal to or greater than the ΣIλI of the adjacency matrix of the graph’s dual.
Conjecture:
.
While this appeared to be the case, we need to test more graphs; a proof does not appear
obvious. In general, it would be interesting to look at what other relationships exist or are
possible between the eigenvalues of planar graphs and their duals.
Another direction future research could take would be to consider the Laplace matrices of
planar graphs and their duals. In the examples we considered we was noted that the main
theorem stipulated in this paper held true for Laplace matrices, except that rather than ΣIλI,
the equation was true for 3Σλ.
References
Marcus, Daniel A. graph theory. A problem oriented approach. MAA Textbooks.
Mathematical Association of America, Washington, DC, 2008. xvi+205 pp. ISBN:
978-0-88385-753-3 MR2503650 (2010h:05002)
Diestel, Reinhard. graph theory. Fourth edition. Graduate Texts in Mathematics, 173.
Springer, Heidelberg, 2010. xviii+437 pp. ISBN: 978-3-642-14278-9
MR2744811 (2011m:05002)
Brouwer, Andries E.; Haemers, Willem H. Spectra of graphs. Universitext. Springer,
New York, 2012. xiv+250 pp. ISBN: 978-1-4614-1938-9 MR2882891 (Reviewed)