spectra of graphs with pendants · spectra of graphs with pendants akito suzuki(shinshu univ.)...
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Spectra of graphs with pendants
Akito Suzuki(Shinshu Univ.)
Related papers:
1. A. Suzuki, Spectrum of the Laplacian on a covering graph with pendant edges I,
Linear Algebra Appl. 439, 3464–3489, 2013.
2. I. Sasaki, A. Suzuki, Essential spectrum of the discrete Laplacian on a deformed lattice,
in preparation.
International Conference
Inverse Problem and Related Topics
18th Aug. 2014
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1. Graphs with pendants
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Graphene: hexagonal lattice consisting of carbon atoms
: A carbon atom 3
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Graphene: hexagonal lattice consisting of carbon atoms
: A carbon atom is regarded as a vertex with a degree of 3.
A chemical bond between
two carbon atoms is regarded
as an edge.
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Graphane (with an a): Fully hydrogenated graphene
: A carbon atom
: A hydrogen atom5
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Graphane (with an a): Fully hydrogenated graphene
: A carbon atom corresponds to a vertex with a degree of 4.
: A hydrogen atom corresponds to a vertex with a degree of 1. → a pendant vertex6
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Graphane (with an a): Fully hydrogenated graphene
: A carbon atom corresponds to a vertex with a degree of 4.
Graphane is a graph with pendants.
: A hydrogen atom corresponds to a vertex with a degree of 1. → a pendant vertex7
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: A carbon atom
: A hydrogen atom
Graphone: Half hydrogenated graphene (triangle type)
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: A carbon atom corresponds to a vertex with a degree of 3 or 4.
: A hydrogen atom corresponds to a pendant vertex.
Graphone: Half hydrogenated graphene (triangle type)
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Graphone: Half hydrogenated graphene (rectangle type)
: A carbon atom corresponds to a vertex with a degree of 3 or 4.
: A hydrogen atom corresponds to a pendant vertex.10
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Graphone: Half hydrogenated graphene (another type)
: A carbon atom corresponds to a vertex with a degree of 3 or 4.
: A hydrogen atom corresponds to a pendant vertex.11
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The Laplacian is defined as follows:
Let be a graph, where and
are the sets of vertices and edges.
where is in the Hilbert space
It is well known that is bounded, self-adjoint and
Spectra of graphs
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Graphene
Graphane
The spectrum of the Laplacian on the graph
that corresponds to graphene is
-1
The spectrum of the discrete Laplacian on
The graph that corresponds to
graphane is
0 21
1
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Graphene
Graphane
The spectrum of the Laplacian on the graph
that corresponds to graphene is
-1
The spectrum of the Laplacian on the graph
that corresponds to graphane is
-1 10
1
A spectral gap opens!
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Triangle type Rectangle type
Another type
Spectral gap
0 - eigenvalue
No spectral gap
Spectral gap
0 21
0 21
0 2
What do you think
the spectra of graphones are?15
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Triangle type Rectangle type
Another type
Spectral gap near 0 and
zero eigenvalue exist
No spectral gap exists
Spectral gap near 0
-1 10
-1 10
-1 1
-1/2 1/2
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Definition: We say that an operator has a spectral gap near zero if
has a spectral gap near zero
has no spectral gap near zero
with some .
What arrangement of pendants makes
a spectral gap near zero and zero eigenvalue?
In this talk, we focus on the following problem:
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2. Preliminary
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The adjacency operator of a graph is
where is a vector in the Hilbert space
Lemma
(1)
(2)
Adjacency operator and Laplacian
From this lemma,
it suffices to know the
spectrum of .
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(1) is proven as follows. Note that
where is a unitary defined by
with
.
We know that
which implies .
Proof of Lemma
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(2) is also proven by
,
which implies
provided ,
and
provided .
Therefore
.
Proof of Lemma
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Assume that have pendant vertices:
Let be an arbitrary set and
.
Suppose that
(H0)
Put
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aExample of a graph satisfying (H0):
(H0))
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aExample of a graph satisfying (H0):
(H0))
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aExample of a graph satisfying (H0):
(H0))
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aExample of a graph satisfying (H0):
(H0))
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aExample of a graph satisfying (H0):
(H0))
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aFor a graph satisfying (H0),
we define a graph by removing vertices of from ,
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aFor a graph satisfying (H0),
we define a graph by removing vertices of from ,
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aaFor a graph satisfying (H0),
we define a graph by removing vertices of from ,
and by removing vertices of from .
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aaFor a graph satisfying (H0),
we define a graph by removing vertices of from ,
and by removing vertices of from .
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Let satisfy (H0). Then can be decomposed into
and can be written as
,
where is defined as
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The following hold:
We have
,
where
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3. Main Results
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Existence of a zero eigenvalue
Let satisfy (H0) and be as above.
Theorem 1. (S.)
In particular, has zero eigenvalue iff. has zero eigenvalue.
Remark:
For bipartite finite graphs, the same statement is found in the book
entitled “Spectra of graphs’’ by Cvetković, Doob and Sachs.
Theorem 1 means that the same statement holds true for infinite graphs
that are not bipartite.k35
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Sketch of proof.
Let . Then
Indeed, we have
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From (H0):
we have .
We have
Hence
and
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Hence we have
.
We can prove that
,
which implies that . 38
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Isospectral transformation
Let be a graph (possibly not satisfying (H0)) and
be as above.
Let be the discrete Schrödinger operator on defined by
where
with
.
Theorem 2. (S.)
and
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Proof of Theorem 2
We use the following well-known lemma.
Lemma. Let
be a bounded self-adjoint operator on with
.
If is bounded, then
is bounded self-adjoint on ,
and
We call the Feshbach map of .40
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Proof of Theorem 2
Applying the above lemma to as
,
the Feshbach map of is
since is bounded provided .
Here we get
.
Hence we have . 41
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Corollary of Theorem 2 (absence of a gap)
Let and be a graph obtained by adding pendant
vertices to .
Corollary 1. Suppose that
.
Then
.
In particular, there is no spectral gap near zero: .
Remark: This corollary can be generalized to and
hexagonal lattice.
Proof of Corollary 1: It suffices to show the spectrum of
includes ,
which comes from .
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Corollary of Theorem 2 (existence of a gap)
Let be a graph obtained by adding pendant vertices to
each vertex of a graph .
Corollary 2. has a spectral gap near zero: .
Proof of Corollary 2: By assumption, .
Let . Then, for sufficiently small,
with some . 43
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Criteria for existence of a gap.
Let be a graph obtained by adding pendant vertices to .
Theorem 3. .
Sketch of proof:
Since , we know that is
bounded if is sufficiently small.
Hence the Feshbach map of
can be defined by
provided . 44
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We can show that
with some independent of .
By assumption, .
If has no zero eigenvalue, then we know that
,
which implies that
.
It the case where has a zero eigenvalue, it can be proven by
employing the Feshbach map of .
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Remark.
For a periodic graph , we can show that
Theorem 4. Suppose that has no zero eigenvalue. Then:
.
This is equivalent to the reverse statement of Theorem 3:
We omit the proof.
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Thank you for your kind attention!
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