Spectrum of the Laplacian of an asymmetric fractal graph

Abstract

We consider a simple self-similar sequence of graphs that does not satisfy the symmetry conditions that imply the existence of a spectral decimation property for the eigenvalues of the graph Laplacians. We show that, for this particular sequence, a very similar property to spectral decimation exists, and we obtain a complete description of the spectra of the graphs in the sequence.

Full text

Spectrum of the Laplacian of an asymmetric
fractal graph
Jonathan Jordan
University of Sheffield ∗
22nd September 2004
Abstract
We consider a simple self-similar sequence of graphs which does not
satisfy the symmetry conditions which imply the existence of a spectral
decimation property for the eigenvalues of the graph Laplacians. We
show that, for this particular sequence, a very similar property to spectral
decimation exists, and obtain a complete description of the spectra of the
graphs in the sequence.
AMS 2000 subject classification: Primary 28A80
1 Introduction and definitions
1.1 Introduction
Many self-similar graphs, and related fractals, display a property known as
spectral decimation, that the spectrum of the Laplacian can be described in
terms of the iteration of a rational function f. Eigenvalues λof the Laplacian
at a given stage of the construction are related to eigenvalues µof the Laplacian
at the following stage of the construction by a relationship
λ=f(µ),(1)
∗Department of Probability and Statistics, University of Sheffield, Hounsfield Road,
Sheffield S3 7RH, U.K.
1

where fis a rational function on R, unless µis a member of a small exceptional
set,E. This was first observed for the specific case of the Sierpi´nski gasket graph
in [7], and this was given a rigorous mathematical treatment in [4, 10, 11]. In the
case of the Sierpi´nski gasket, using our definition of the Laplacian (see section
1.3), the function f(µ) = µ(5 −4µ) and the exceptional set is {1/2,5/4,3/2}.
A generalisation of spectral decimation to a much larger class of self-similar
graphs, including the Vicsek set graph, appears in [6], in which a symmetry
condition is developed which, if satisfied, ensures that spectral decimation ap-
plies to the graph. Each self-similar graph in this class has a function fand
exceptional set Eassociated with it.
In this paper we consider a simple asymmetric self-similar graph which does not
satisfy the symmetry condition of [6]. In section 1.2 we define a sequence of
graphs (Gn)n∈N, which can be used to define a self-similar graph G∞as for the
self-similar graphs in [6].
In section 2, we show that, for this example, a property similar to spectral
decimation exists, in which (1) is replaced by
2(1 −λ)2=f(µ) (2)
where f(µ) is a quartic polynomial. Again there is an exceptional set of values of
µwhere the relationship does not necessarily hold. This is proved in Theorems
1 and 2.
Another common spectral property of self-similar graphs and related fractals
is that there are many eigenvalues of the Laplacian with high multiplicity and
Dirichlet-Neumann eigenfunctions, i.e. eigenfunctions which are zero on the
boundary. In [8] it is shown that the eigenvalues with Dirichlet-Neumann eigen-
functions dominate the spectrum in a large class of cases, that of the nested
fractals introduced in [5].
In section 3, we calculate the number of linearly independent eigenfunctions of
the Laplacian which are Dirichlet-Neumann or non-Dirichlet-Neumann, showing
that for large nthe spectrum of the Laplacian of Gnis dominated by eigenvalues
with Dirichlet-Neumann eigenfunctions. Together with the relationship between
the eigenvalues this allows us, in section 4, to describe the spectra of the graphs
in the self-similar sequence of finite graphs used in the construction of our graph.
This is stated in Theorem 6, which gives a complete description of the spectrum,
including the multiplicity of the eigenvalues and which eigenvalues are associated
with Dirichlet-Neumann and non-Dirichlet-Neumann eigenfunctions.
An example of a self-similar graph which does not satisfy the symmetry condi-
tions of [6], and for which spectral decimation appears not to apply, is associated
2

with the pentagasket, as described in [1], in which numerical approximations for
eigenvalues and eigenvectors are obtained, and some theoretical results are ob-
tained, showing how to construct eigenspaces of high multiplicity.
A more complicated method, using a rational map on a projective variety rather
than on R, which works for a larger class of self-similar graphs than that in [6]
including some for which spectral decimation does not apply, is described in [9].
However, our graph does not meet all the conditions described in section 1.1.1
of [9].
1.2 The graph
We define a self-similar sequence of finite graphs (Gn)n∈N, with V(Gn−1)⊆
V(Gn).
We start with G0, a single edge between two vertices 1 and 2. To construct
Gn+1, we note that each edge eof Gnconnects two vertices i(e) and j(e),
i(e)∈V(Gn−1) and j(e)∈V(Gn)\V(Gn−1). For each eof Gnwe obtain a
new vertex k(e)∈V(Gn+1), and the edges of Gn+1 consist of, for each edge e
of Gn, 2 edges connecting i(e) and k(e) and one edge connecting j(e) and k(e).
The following pictures show G1and G2:
} } }
xxxx
x
x
3

The same sequence of graphs can be obtained by using the framework of Def-
inition 5.2 of [6], with the model graph being identical to G1above, but with
conditions on the orientation to deal with the asymmetry.
When 1 ≤m < n, the graph Gncontains 3n−msubgraphs isomorphic to Gm.
We will call these subgraphs m-cells. Using this, we can define a sequence
(˜
Gn)n∈Nsuch that ˜
Gnis isomorphic to Gnand ˜
Gmis a subgraph of ˜
Gnfor
m < n. We then define the infinite graph G∞=Sn=0 ∞˜
Gn. This is analogous
to the Definition 5.5 of [6].
We define maps fi:V(Gn−1)→V(Gn), i = 1,2,3 mapping each vertex of Gn−1
to the corresponding vertex in each (n−1)-cell. We will label these so that f1
and f2correspond to the two parallel cells.
We note that this graph is similar to that described in [3], although in the
context of that paper the orientation of the cells is not important.
The graph Gnis bipartite with the two parts being V(Gn−1) and V(Gn)\
V(Gn−1).
1.3 The Laplacian
There are a number of different definitions of the Laplacian of a graph. The
definition of the graph Laplacian used in [6] is the generator matrix of a contin-
uous time random walk on the graph, while in the book [2] a related symmetric
matrix is used. However the eigenvalues of the different definitions differ by at
most a simple transformation.
For convenience in describing the eigenfunctions, we use the following definition
of the Laplacian: the Laplacian LGof a graph (which may have multiple edges
but with no loops) Gis a |V(G)|×|V(G)|matrix with, for a vertex i∈V(G),
LG(i, i) = 1, and, for i, j ∈V(G) with i6=j,LG(i, j ) = −ei,j
δi, where ei,j is
the number of edges linking iand jin Gand δiis the degree of vertex iin G.
This gives the same eigenvalues as the symmetric Laplacian described in [2],
and the eigenfunctions are the “harmonic eigenfunctions” described in [2]. Our
definition of the graph Laplacian differs from that in [6] only in that the sign of
each entry (and hence of the eigenvalues) is reversed.
4

2 The relationship between the eigenvalues
We set f(µ) = 9(µ−1)4−9(µ−1)2+ 2, so that (2) becomes
2(1 −λ)2= 9(µ−1)4−9(µ−1)2+ 2.(3)
We first show how to construct eigenvalues µof LGn+1 from eigenvalues λof
LGnwhen λ /∈ {0,1,2}.
Given λand µ, we set
γ=3(µ−1)2−2
1−λ.(4)
Theorem 1. Given an eigenfunction xof LGnwith eigenvalue λ /∈ {0,1,2},
we can
•Solve (3) for µto obtain 4 roots
•For each possible µ, set γusing (4).
•Then define x0by
x0
i=½xii∈V(Gn−1)
γxii∈V(Gn)\V(Gn−1)(5)
and, for a vertex k=k(e)∈V(Gn+1)\V(Gn), we set
x0
k=2xi(e)
3(1 −µ)+γxj(e)
3(1 −µ).(6)
Then x0is an eigenfunction of LGn+1 with eigenvalue µ.
Proof. To check this, we just calculate LGn+1x0. For i∈V(Gn−1),
(LGn+1 x0)i=xi+1
δ(n)
iX
e∈E(n)
iµ2xi
3(µ−1) +γxj(e)
3(µ−1)¶
=xi+2xi
3(µ−1) +γxi(1 −λ)
3(µ−1)
=xiµ3(µ−1) + 2 + 3(µ−1)2−2
3(µ−1) ¶
=µxi=µx0
i.
5

For j∈V(Gn)\V(Gn−1),
(LGn+1 x0)j=γxj+1
δ(n)
jX
e∈E(n)
jµ2xi(e)
3(µ−1) +γxj
3(µ−1)¶
=γxj+γxj
3(µ−1) +2xj(1 −λ)
3(µ−1)
=xjµ3(µ−1)γ+γ+ 2(1 −λ)
3(µ−1) ¶
Using (3) and (4),
2(1 −λ) = 9(µ−1)4−9(µ−1)2+ 2
1−λ
=(3(µ−1)2−1)(3(µ−1)2−2)
1−λ
= (3(µ−1)2−1)γ
and so
(LGn+1 x0)j=xjγµ3(µ−1)+1+3(µ−1)2−1
3(µ−1) ¶
=µγxj
=µx0
j
and finally, for k∈j∈V(Gn)\V(Gn−1), which satisfies k=k(e) for some edge
eof Gn, we have
(LGn+1 x0)k=x0
k−2
3x0
i(e)−1
3x0
j(e)
=µ1
1−µ−1¶µ2
3xi+γ
3xj¶
=µµ2xi
3(1 −µ)+γxj
3(1 −µ)¶
=µx0
k,
so x0is indeed an eigenfunction of LGn+1 with eigenvalue µ.
Theorem 2. If µ /∈ {1,1 + p2/3,1−p2/3,1 + p1/3,1−p1/3}and λand
µsatisfy (3), then µis an eigenvalue of Gn+1 if and only if λis an eigenvalue
of Gn, with the same multiplicity.
Proof. If we have an eigenfunction x0of LGn+1 with eigenvalue µ6= 1, then, for
each edge eof Gn+1,
x0
k(e)=2x0
i(e)
3(1 −µ)+x0
j(e)
3(1 −µ)
6

so that, for each i∈V(Gn−1),
x0
i(1 −µ) = 1
δ(n)
iX
e∈E(n)
i
Ã2x0
i
3(1 −µ)+x0
j(e)
3(1 −µ)!
=2x0
i
3(1 −µ)+1
δ(n)
iX
e∈E(n)
i
x0
j(e)
3(1 −µ)
giving
x0
i(3(1 −µ)2−2) = 1
δ(n)
iX
e∈E(n)
i
x0
j(e).(7)
Similarly for j∈V(Gn)\V(Gn−1),
x0
j(1 −µ) = 1
δ(n)
jX
e∈E(n)
j
Ãx0
j
3(1 −µ)+2x0
i(e)
3(1 −µ)!
=x0
j
3(1 −µ)+1
δ(n)
jX
e∈E(n)
j
2x0
i(e)
3(1 −µ)
giving
x0
j(3(1 −µ)2−1) = 1
δ(n)
jX
e∈E(n)
j
x0
i(e).(8)
By the conditions of the theorem (1 −µ)26=2
3. Then (7) implies that, for any
λ,
x0
i(1 −λ) = 1
δ(n)
iX
e∈E(n)
i
1−λ
3(1 −µ)2−2x0
j(e)(9)
while (8) gives, if λ6= 1,
1−λ
3(1 −µ)2−2x0
j
(3(1 −µ)2−2)(3(1 −µ)2−1)
2(1 −λ)=1
δ(n)
jX
e∈E(n)
j
x0
i(e).(10)
Then, if we set xi=x0
ifor i∈V(Gn−1) and
xj=x0
j
1−λ
3(1 −µ)2−2,
(9) and (10) become
xi(1 −λ) = 1
δ(n)
iX
e∈E(n)
i
xj(e)
7

and
xj
(3(1 −µ)2−2)(3(1 −µ)2−1)
2(1 −λ)=1
δ(n)
jX
e∈E(n)
j
xi(e)
which imply that xis an eigenfunction of LGnwith eigenvalue λif
(1 −λ) = (3(1 −µ)2−2)(3(1 −µ)2−1)
2(1 −λ),
which is equivalent to the quartic (3).
This eigenfunction can be degenerate only if 1−λ= 0, i.e. if either (1−µ)2= 1/3
or (1 −µ)2= 2/3.
The set {1,1 + p2/3,1−p2/3,1 + p1/3,1−p1/3}of values of µwhere
Theorem 2 does not apply plays a similar role to that of the exceptional set in
[6].
We note that the eigenvalues λand 2 −λproduce the same values of µ, with
the same eigenfunctions. This is related to the bipartite nature of the graph
- in fact if xis an eigenfunction with eigenvalue λthen, following [2], we can
obtain an eigenfunction with eigenvalue 2 −λby simply changing the sign of
xon V(Gn)\V(Gn−1). These two eigenfunctions will then produce the same
new eigenfunction using the above construction.
We now consider the cases when Theorems 1 and 2 do not apply, i.e. when
λ∈ {0,1,2}or µ∈ {1,1 + p2/3,1−p2/3,1 + p1/3,1−p1/3}.
We note that if µ= 1 and λand µsatisfy (3), then λ= 0 or 2.
When λ= 1 and xi6= 0 for some i∈V(Gn−1), we use the same method, but
with γ= 0 and the quartic (3) replaced by
(µ−1)2= 2/3.(11)
We cannot use this method if xi= 0 for all i∈V(Gn−1), because the con-
structed eigenfunction would be zero everywhere.
However, in the case where λ= 1 and xi= 0 for all i∈V(Gn−1), we can
construct an eigenfunction x0by setting x0
i= 0 for i∈V(Gn−1), and x0
i=xi
for i∈V(Gn)\V(Gn−1). This gives eigenvalues µwith
(µ−1)2= 1/3,(12)
using similar methods to those above.
8

3 Dirichlet-Neumann and non-Dirichlet-Neumann
eigenfunctions
The graph Gnhas vnvertices and enedges where v0= 2, e0= 1 and en= 3en−1,
vn=vn−1+en−1. Hence en= 3nand vn=1
2(3n+ 3).
The following lemma provides a means of constructing Dirichlet-Neumann eigen-
functions, which are zero on the two boundary vertices 1 and 2.
Lemma 3. Let G0be a connected graph with mvertices, including distinguished
endpoints 1and 2, and let Gbe the graph formed by defining G1and G2to be two
identical copies of G0and connecting them in parallel by identifying their end-
points. Then the Laplacian of Ghas m−2linearly independent eigenfunctions
which are zero on the endpoints.
The associated eigenvalues are the eigenvalues of the Laplacian LGrestricted to
the set {2j: 2 ≤j≤m−1}of vertices in G2.
Proof. We label the vertices of G01,2, . . . , m. Then we label the vertices in G
so that, for j≥3, vertex jin G0corresponds to vertex 2j−3 in G1and vertex
2j−2 in G2.
Now consider the Laplacian LG. For 2 ≤j, k ≤m−1 we have
LG(1,2j−1) = LG(1,2j)
LG(2,2j−1) = LG(2,2j)
LG(2j−1,2k−1) = LG(2j, 2k)
LG(2j−1,2k) = LG(2j, 2k−1) = 0
and consider functions xsatisfying
x(1) = x(2) = 0
x(2j−1) = −x(2j) for 2 ≤j≤m−1.
Now
(LGx)(1) =
m−1
X
j=2
(LG(1,2j−1)x(2j−1) + LG(1,2j)x(2j)) = 0
9

and similarly (LGx)(2) = 0, while
(LGx)(2j−1) =
m−1
X
k=2 LG(2j−1,2k−1)x(2k−1)
=−
m−1
X
k=2 LG(2j, 2k)x(2k) = (LGx)(2j).
So the Laplacian LGpreserves vectors of this form, which form a vector space
of dimension m−2, and it acts on them like the Laplacian restricted to the
interior vertices of G2. As the Laplacian is symmetric, there are m−2 linearly
independent Dirichlet-Neumann eigenfunctions of the Laplacian.
If Gncontains a subgraph Gof this form, where the vertices of Gother than
the endpoints have no edges linking them to Gn\G, then we can take one of
the eigenfunctions xon Gconstructed by the above lemma and extend it to an
eigenfunction ˜xon Gnby setting
˜x(v) = x(v) for v∈V(G)
˜x(v) = 0 otherwise
If the endpoints 1 and 2 are both not in the interior of the subgraph G, then
this ˜xwill be Dirichlet-Neumann.
Proposition 4. The graph Gnhas at least 1
2(3n+3)−2n−1linearly independent
Dirichlet-Neumann eigenvalues.
Proof. The model graph contains parallel edges, so Gncontains a subgraph
consisting of two copies of Gn−1with their boundary points identified as in
Lemma 3. This gives vn−1−2 eigenfunctions. For each eigenfunction xobtained
thus, we have x(f1(v)) = −x(f2(v)) and x(f3(v)) = 0 for each v∈V(Gn−1).
Furthermore, given a Dirichlet-Neumann eigenfunction xof Gn−1we can obtain
3 Dirichlet-Neumann eigenfunctions x1, x2, x3of Gnby extending them from
(n−1)-cells to the whole graph i.e.
xi(fj(v)) = δij x(v) for all v∈V(Gn−1), i, j = 1,2,3.
However, we only obtain 2 linearly independent eigenfunctions, because the
linear combination x1−x2is of the form obtained using Lemma 3.
Hence, if lnis the number of Dirichlet-Neumann eigenvalues constructed by
these methods, it satisfies
ln= 2ln+1
2(3n−1+ 3) −2
and l2= 1, which gives the result.
10

Let the total number of Dirichlet-Neumann eigenfunctions of Gnbe ln+ˆ
ln, so
that ˆ
lnis the number which are not constructed by the methods of Proposition
4
For this graph, we can also describe a set of non-Dirichlet-Neumann eigenfunc-
tions.
Proposition 5. The graph Gnhas 2n+ 1 linearly independent eigenfunctions
which are not Dirichlet-Neumann.
Proof. We consider the set of eigenfunctions xwhich are zero on the central
vertex of V(G1), i.e. they satisfy x(3) = 0 and, for each v∈V(Gn−1),
x(f1(v)) = x(f2(v)) = 1
2x(f3(v)), a (vn−1−1)-dimensional subspace. This
is preserved by the Laplacian of Gn, so we can find vn−1−1 = 1
2(3n−1+ 1)
linearly independent eigenfunctions satisfying these properties.
To exclude those which are Dirichlet-Neumann, this is equivalent to the con-
dition that x(1) = x(2) = 0. So a Dirichlet-Neumann eigenfunction satisfying
the above conditions reduces to a Dirichlet-Neumann eigenfunction on each
(n−1)-cell. Hence there are 1
2(3n−1+ 1) −ln−1−ˆ
ln−1= 2n−1−ˆ
ln−1non-
Dirichlet-Neumann eigenfunctions satisfying the above conditions.
Now, given any eigenfunction xof Gn−1, we can extend it to an eigenfunction
x0of Gnby setting x0(1) = √2x(1), x0(2) = x(1), x0(3) = √3x(2), x0(f1(v)) =
x0(f2(v)) = x0(f3(v)) = x(v) for v≥3, from the structure of the graph. This
means that each of the eigenfunctions constructed in Proposition 5 for Gn−1
can be extended to a non-Dirichlet-Neumann eigenfunction of Gn, which will be
linearly independent of those already found (because x0(3) 6= 0). Inductively,
this also applies to those constructed for Gn−m, m > 1.
The graphs are bipartite, so eigenfunctions (which are zero nowhere) for eigen-
values 0 and 2 exist as described in [2].
Hence the total number of linearly independent non-Dirichlet-Neumann eigen-
functions is at least 2n+ 1 −Pn−1
m=0 ˆ
ln. But we know that there are exactly
1
2(3n+ 3) linearly independent eigenfunctions. Hence
1
2(3n+ 3) ≥2n+ 1 −
n−1
X
m=0
ˆ
lm+1
2(3n+ 3) −2n−1 + ˆ
ln
and so ˆ
ln≤Pn−1
m=0 ˆ
lm. But ˆ
lm= 0 for m≤2, and hence for all m.
Hence the eigenfunctions constructed are all that exist, and there are 2n+ 1 lin-
early independent non-Dirichlet-Neumann ones, the remainder being Dirichlet-
Neumann.
11

4 The spectra of the graphs
We can now use the relationships between eigenvalues and the information on
Dirichlet-Neumann and non-Dirichlet-Neumann eigenfunctions to obtain a com-
plete description of the spectra of the graphs Gn.
Theorem 6. Set α(1)
1= 1,α(2)
1= 1 −q2
3and α(2)
2= 1 + q2
3. We extend this
to define {α(n)
i; 1 ≤i≤2n−1}to be the 2n−1values µsatisfying the quartic (3),
with λ=α(n−1)
jfor some j.
Similarly set β(1)
1= 1,β(2)
1= 1 −q1
3and β(2)
2= 1 + q1
3. We extend this to
define {β(n)
i; 1 ≤i≤2n−1}to be the 2n−1values µsatisfying the quartic (3),
with λ=β(n−1)
jfor some j.
Then
(a) If n≥m, then LGnhas a non-Dirichlet-Neumann eigenfunction with
eigenvalue α(m)
i,1≤i≤2m−1. Together with eigenfunctions with eigen-
values 0and 2, this describes the non-Dirichlet-Neumann spectrum of LGn.
(b) If n≥m+ 1, then LGnhas 1
2(3n−m−1) linearly independent Dirichlet-
Neumann eigenvalues with eigenvalue β(m)
i, for 1≤i≤2m−1.
Proof. We note that G1has a non-Dirichlet-Neumann eigenfunction with eigen-
value 1. As described in the proof of Proposition 5, this can then be extended
to give a non-Dirichlet-Neumann eigenfunction xwith eigenvalue 1 for each Gn,
n≥1.
Because this eigenfunction is non-Dirichlet-Neumann, it has xi6= 0 for at least
some i∈V(Gn−1). Hence the construction of eigenfunctions of Gn+1 with
eigenvalues µsatisfying (1 −µ)2=2
3produces non-degenerate eigenfunctions,
which are also non-Dirichlet-Neumann. So Gn+1 has non-Dirichlet-Neumann
eigenfunctions with eigenvalues 1 ±q2
3.
We have already shown that Gnhas a non-Dirichlet-Neumann eigenfunction
with eigenvalue α(2)
i, 1 ≤i≤2n−1, for n≥2. Now, if Gn−1has a non-Dirichlet-
Neumann eigenfunction with eigenvalue α(m−1)
ifor each 1 ≤i≤2m−2, then the
construction of eigenfunctions gives us a non-Dirichlet-Neumann eigenfunction
of Gnwith eigenvalue α(m)
ifor each 1 ≤i≤2m−1. Using this inductively, we
12

find that Gnhas a non-Dirichlet-Neumann eigenfunction with eigenvalue α(m)
i,
1≤i≤2m−1, for n≥m.
Along with the eigenfunctions with eigenvalues 0 and 2, this gives us all 2n+ 1
non-Dirichlet-Neumann eigenfunctions from Proposition 5, and hence completes
the proof of (a).
We now consider the Dirichlet-Neumann eigenfunctions. We note that G2has
a Dirichlet-Neumann eigenfunction with eigenvalue 1. Such an eigenfunction
is zero on V(G1), so our main construction produces a degenerate eigenfunc-
tion. However the alternative construction with (1 −µ)2=1
3does produce two
eigenfunctions of G3. Hence, as there are 1
2(3n+ 3) −2n−1 Dirichlet-Neumann
eigenfunctions of Gn, we can use the constructions to obtain 3n+ 3 −2n+1 −2
Dirichlet-Neumann eigenfunctions, with eigenvalues other than 1, of Gn+1.
We now show that, for n≥2, Gnhas 1
2(3n−1−1) linearly independent Dirichlet-
Neumann eigenfunctions with eigenvalue 1. This is the case for n= 2. For each
such eigenfunction of Gn, we can construct 3 of Gn+1 using the methods in the
proof of Proposition 4.
Assuming that Gnhas 1
2(3n+3)−2n−1 linearly independent Dirichlet-Neumann
eigenfunctions of which 1
2(3n−1−1) have eigenvalue 1, we have 1
2(3n−3) linearly
independent Dirichlet-Neumann eigenfunctions of Gn+1 with eigenvalue 1, and
3n+ 3 −2n+1 −2 with other eigenvalues. However, we know from Proposition 4
that there are 1
2(3n+1 +3)−2n+1 −1 in total, and 1
2(3n−3)+3n+ 3 −2n+1 −2 =
1
2(3n+1 + 3) −2n+1 −2.
The one unexplained eigenfunction must also have eigenvalue 1, because if it had
eigenvalue λ6= 1 we would also have an unexplained eigenfunction with eigen-
value 2 −λ. Hence we have 1
2(3n−1) linearly independent Dirichlet-Neumann
eigenfunctions of Gn+1 with eigenvalue 1, giving the result by induction.
We know that, for n≥2, Gnhas 1
2(3n−1−1) linearly independent Dirichlet-
Neumann eigenfunctions with eigenvalue 1. We now use our constructions m
times to show that, for n≥m+ 1, Gnhas 1
2(3n−m−1) linearly independent
Dirichlet-Neumann eigenfunctions with eigenvalue β(m)
i, for 1 ≤i≤2m−1. This
completes the proof of (b).
13

5 Reversing the orientation
We remark that very similar results can be obtained if we reverse the orientation
of the model graphs in the definitions of Section 1.2. Because of the asymmetry
this gives a different self-similar sequence of graphs. Eigenvalues λand µof
Laplacians successive members of the sequence are related by the same equation
(3) when λ /∈ {0,1,2}, but the two equations (11) and (12) for µwhen λ= 1
are reversed. This has the effect that, in Theorem 6, the roles of α(n)
iand β(n)
i
are reversed.
References
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