10,763 research outputs found
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
It is well known that the spectral radius of a tree whose maximum degree is
cannot exceed . In this paper we derive similar bounds for
arbitrary planar graphs and for graphs of bounded genus. It is proved that a
the spectral radius of a planar graph of maximum vertex degree
satisfies . This result is
best possible up to the additive constant--we construct an (infinite) planar
graph of maximum degree , whose spectral radius is . This
generalizes and improves several previous results and solves an open problem
proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus.
For every , these bounds can be improved by excluding as a
subgraph. In particular, the upper bound is strengthened for 5-connected
graphs. All our results hold for finite as well as for infinite graphs.
At the end we enhance the graph decomposition method introduced in the first
part of the paper and apply it to tessellations of the hyperbolic plane. We
derive bounds on the spectral radius that are close to the true value, and even
in the simplest case of regular tessellations of type we derive an
essential improvement over known results, obtaining exact estimates in the
first order term and non-trivial estimates for the second order asymptotics
Nodal count of graph eigenfunctions via magnetic perturbation
We establish a connection between the stability of an eigenvalue under a
magnetic perturbation and the number of zeros of the corresponding
eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a
graph and count the number of edges where the eigenfunction changes sign (has a
"zero"). It is known that the -th eigenfunction has such zeros,
where the "nodal surplus" is an integer between 0 and the number of cycles
on the graph.
We then perturb the Laplacian by a weak magnetic field and view the -th
eigenvalue as a function of the perturbation. It is shown that this function
has a critical point at the zero field and that the Morse index of the critical
point is equal to the nodal surplus of the -th eigenfunction of the
unperturbed graph.Comment: 18 pages, 4 figure
Supersymmetry on Graphs and Networks
We show that graphs, networks and other related discrete model systems carry
a natural supersymmetric structure, which, apart from its conceptual importance
as to possible physical applications, allows to derive a series of spectral
properties for a class of graph operators which typically encode relevant graph
characteristics.Comment: 11 pages, Latex, no figures, remark 4.1 added, slight alterations in
lemma 5.3, a more detailed discussion at beginning of sect.6 (zero
eigenspace
Multiport Impedance Quantization
With the increase of complexity and coherence of superconducting systems made
using the principles of circuit quantum electrodynamics, more accurate methods
are needed for the characterization, analysis and optimization of these quantum
processors. Here we introduce a new method of modelling that can be applied to
superconducting structures involving multiple Josephson junctions, high-Q
superconducting cavities, external ports, and voltage sources. Our technique,
an extension of our previous work on single-port structures [1], permits the
derivation of system Hamiltonians that are capable of representing every
feature of the physical system over a wide frequency band and the computation
of T1 times for qubits. We begin with a black box model of the linear and
passive part of the system. Its response is given by its multiport impedance
function Zsim(w), which can be obtained using a finite-element electormagnetics
simulator. The ports of this black box are defined by the terminal pairs of
Josephson junctions, voltage sources, and 50 Ohm connectors to high-frequency
lines. We fit Zsim(w) to a positive-real (PR) multiport impedance matrix Z(s),
a function of the complex Laplace variable s. We then use state-space
techniques to synthesize a finite electric circuit admitting exactly the same
impedance Z(s) across its ports; the PR property ensures the existence of this
finite physical circuit. We compare the performance of state-space algorithms
to classical frequency domain methods, justifying their superiority in
numerical stability. The Hamiltonian of the multiport model circuit is obtained
by using existing lumped element circuit quantization formalisms [2, 3]. Due to
the presence of ideal transformers in the model circuit, these quantization
methods must be extended, requiring the introduction of an extension of the
Kirchhoff voltage and current laws
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