# Spectral estimates for infinite quantum graphs

@article{Kostenko2018SpectralEF, title={Spectral estimates for infinite quantum graphs}, author={Aleksey Kostenko and Noema Nicolussi}, journal={Calculus of Variations and Partial Differential Equations}, year={2018}, volume={58}, pages={1-40} }

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of…

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## References

SHOWING 1-10 OF 71 REFERENCES

Spectral Theory of Infinite Quantum Graphs

- MathematicsAnnales Henri Poincaré
- 2018

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths…

On the Spectral Gap of a Quantum Graph

- Mathematics
- 2015

We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices,…

Infinite quantum graphs

- Mathematics
- 2017

Infinite quantum graphs with δ-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a…

Edge connectivity and the spectral gap of combinatorial and quantum graphs

- Mathematics, Computer Science
- 2017

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of Laplacians on combinatorial and quantum graph in terms of the edge connectivity, i.e. the minimal number of edges…

Eigenvalue estimates on quantum graphs

- Mathematics
- 2016

On a finite connected metric graph, we establish upper bounds for the eigenvalues of the Laplacian. These bounds depend on the length, the Betti number, and the number of pendant vertices. For trees,…

Quantum Graphs which Optimize the Spectral Gap

- Mathematics
- 2016

A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of…

Spectral analysis of metric graphs and related spaces

- Mathematics
- 2007

The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs…

Some relations between analytic and geometric properties of infinite graphs

- MathematicsDiscret. Math.
- 1991

Volume growth, spectrum and stochastic completeness of infinite graphs

- Mathematics
- 2011

We study the connections between volume growth, spectral properties and stochastic completeness of locally finite weighted graphs. For a class of graphs with a very weak spherical symmetry we give a…