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## GENERALISED DISCRETE LAPLACIANS ON GRAPHS AND THEIR RELATION TO QUANTUM GRAPHS (2009)

### Citations

3070 | Perturbation Theory for Linear Operators - Kato - 1966 |

219 | Kirchhoff’s rule for quantum wires - Kostrykin, Schrader - 1999 |

181 | L2-invariants: theory and applications to geometry and K-theory - Lück - 2002 |

96 | A survey on spectra of infinite graphs - Mohar, Woess - 1989 |

78 | Periodic orbit theory and spectral statistics for quantum graphs - Kottos, Smilansky - 1999 |

68 | Hermitian symplectic geometry and extension theory - Harmer |

51 | Quantum graphs I. Some basic structures, Waves Random Media 14 - Kuchment - 2004 |

39 | A characteristic equation associated to an eigenvalue problem on c2-networks - Below - 1985 |

38 |
Boundary value problems for elliptic partial differential operators on bounded domains
- Behrndt, Langer
(Show Context)
Citation Context ...ctive, but have dense range only. This is e.g. the case if A is an elliptic differential operator like the Laplacian on a manifold with boundary. For a related concept and more references we refer to =-=[BeL07]-=-. Boundary triples associated to a quadratic form as introduced here are called bounded, elliptic boundary triples associated to a quadratic form in [P09, Sec. 3.4]. We end this section with the const... |

38 | Harmonic analysis on metrized graphs - Baker, Rumely |

36 | Laplacians on metric graphs: Eigenvalues, resolvents and semigroups - Kostrykin, Schrader - 2005 |

33 | Can one hear the shape of a graph - Gutkin, Smilansky - 2001 |

32 | de Verdière: Spectres de graphes - Colin - 1998 |

32 | Quantum wires with magnetic fluxes - Kostrykin, Schrader |

31 | Metrized graphs, Laplacian operators, and electrical networks, Quantum graphs and their applications - Baker, Faber - 2006 |

31 | Heat kernels on metric graphs and a trace formula,” Spectral determinants on graphs 16 - Kostrykin, Potthoff, et al. |

29 |
The spectrum of the continuous Laplacian on a graph.Monatsh
- Cattaneo
- 1997
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Citation Context ...elation for equilateral graphs. The spectral relation between the metric and combinatorial operator for the standard vertex space is well-known, see for example [vB85, Nic87] for the compact case and =-=[Ca97]-=- for the non-compact case (see also [Kuc04, Pan06, P08, BGP08] and the references therein). Moreover, in [Exn97], deltaand delta’-vertex conditions are considered. Dekoninck and Nicaise [DN00] proved ... |

28 | Upper bounds for eigenvalues of the discrete and continuous Laplace operators - Chung, Grigoryan, et al. - 1996 |

28 | Inverse spectral problem for quantum graphs
- Kurasov, Nowaczyk
- 2005
(Show Context)
Citation Context ... structure of the graph is uniquely determined. Further extensions are given e.g. in [KPS07b]. Some results can be extended to the case of (trivially or weakly) rationally dependent edge lengths (see =-=[Now07]-=-), but counterexamples in [R84, GS01, BSS06] show that one needs some conditions on the edge lengths. In particular, there are isospectral, non-homeomorphic graphs. The proof of Theorem 7.5 uses the e... |

27 | Spectra of Schrödinger operators on equilateral quantum graphs - Pankrashkin - 2006 |

26 |
spectre du laplacien sur un graphe, Théorie du potentiel
- Le
- 1983
(Show Context)
Citation Context ...s on metric and discrete graphs Let us finish this chapter with some results concerning the trace of the heat operator. Trace formulas for metric graph Laplacians appeared first in an article of Roth =-=[R84]-=-, where standard (Kirchhoff) boundary conditions are used. Independently, Nicaise proved trace formulas for metric graphs in [Nic87], but he uses a slightly different definition of the Laplacian (as i... |

21 | Quantum graphs: a simple model for chaotic scattering - Kottos, Smilansky |

21 | Spectral convergence of quasi-one-dimensional spaces - Post |

19 | Asymptotics of spectra of Neumann Laplacians in thin domains - Kuchment, Zeng |

15 | Cantor and band spectra for periodic quantum graphs with magnetic fields - Brüning, Geyler, et al. |

14 | Quantum Chaos on graphs Phys - Kottos, Smilansky - 1997 |

14 | Graph laplacians and topology - Kurasov |

12 | Nodal domains on isospectral quantum graphs: the resolution of isospectrality - Band, Shapira, et al. - 2006 |

10 | Exponential localization for radial random quantum trees, Preprint math-ph/0611022 - Hislop, Post - 2006 |

8 | The spectrum of the averaging operator on a network (metric graph
- Cartwright, Woess
(Show Context)
Citation Context ...eferences therein). Moreover, in [Exn97], deltaand delta’-vertex conditions are considered. Dekoninck and Nicaise [DN00] proved spectral relations for fourth order operators, and Cartwright and Woess =-=[CW05]-=- used integral operators on the edge. Let us combine the concrete information on the boundary triple (Γ, Γ ′ , V ) with Theorem A.5, in order to obtain a spectral relation between the quantum and disc... |

7 | Eigenvalue bracketing for discrete and metric graphs
- Lledó, Post
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Citation Context ...se. Using eigenvalue monotonicity w.r.t. the vertex space, one can ensure spectral gaps for the discrete Laplacian on an infinite covering of a finite graph with residually finite covering group, see =-=[LP08]-=- for details. Let us now compare the extended Laplacian ˆ ∆ (V ,L) with the corresponding discrete operators as in Theorem A.12. The Dirichlet solution and the Dirichlet-to-Neumann operator for the bo... |

7 |
Approche spectrale des problemes de diffusion sur les réseaux
- Nicaise
- 1987
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Citation Context ...ulas for metric graph Laplacians appeared first in an article of Roth [R84], where standard (Kirchhoff) boundary conditions are used. Independently, Nicaise proved trace formulas for metric graphs in =-=[Nic87]-=-, but he uses a slightly different definition of the Laplacian (as in [Ca97]). More general self-adjoint vertex conditions (energy-independent, see Remark 3.16 (ii)) are treated in [KS06, KPS07b]. Tra... |

6 |
order approach and index theorems for discrete and metric graphs, Preprint arXiv:0708.3707
- First
- 2007
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Citation Context ...les are borrowed from the corresonding examples in the metric graph case, see the end of Section 3. For more general cases defined via vertex spaces, e.g. the discrete magnetic Laplacian, we refer to =-=[P07b]-=-. (i) Choosing Vv = C (v) = C(1, . . ., 1), we obtain the standard vertex space denoted by V std v , also called continuous or Kirchhoff . The associated projection is Pv = 1 deg v E where E denotes t... |

5 |
equations for graphs and the edge-based Laplacian
- Wave
(Show Context)
Citation Context ... ‖Ψ0F ‖ gives an equivalent norm turning Ψ0 into an isometry. Moreover, ‖Ψ1η‖ = ‖η‖ℓ2 (E). For more details on this point of view (as well as “mixed” types of discrete and metric graphs), we refer to =-=[FT04b]-=- and references therein. Finally, we analyse the spectrum at the bottom of the extended model. Let (G, V ) be a quantum graph and L = L ∗ be a local, bounded operator on V . We define the Hilbert chai... |

3 | A remark on the spectrum of magnetic Laplacianon a graph - Higuchi, Shirai - 1999 |

3 |
analysis of metric graphs and related spaces, Preprint arXiv:0712.1507
- Spectral
- 2007
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Citation Context ...2,v. We call a vertex space Vv without such a decomposition irreducible. Similarly, we say that V = ⊕ v Vv is irreducible, if all its local subspaces Vv are irreducible. For more details, we refer to =-=[P07c]-=-. In [P07b, Lem. 2.13] we showed the following result on symmetry of a vertex space: Proposition 2.10. Assume that the vertex space Vv of a vertex v with degree d = deg v is invariant under permutatio... |

2 |
of self-adjoint extensions and applications to solvable Schrödinger operators
- Spectra
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Citation Context ... Boundary triples Boundary triples allow to express boundary value problems in an purely operatortheoretic way. In this section, we briefly describe this concept, and closely follow the exposition in =-=[BGP08]-=-. For more details and a historical account including more references, we refer to [BGP08, DHMdS06]. In this section, we assume that A is a closed operator in a Hilbert space H such that A ∗ is symmet... |

2 | graphs: II. Some spectral properties of quantum and combinatorial graphs - Quantum |

1 | Snoo, Boundary relations and their Weyl - Derkach, Hassi, et al. |

1 |
kernels on metric graphs and a trace formula
- Heat
- 2007
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Citation Context ...f the metric graph Gmet with vertex space V , i.e., the Fredholm index of dV . In Theorem 6.5 we showed ... that the index is the same as the discrete index ind(G, V ) (the Fredholm index of dV ). In =-=[KPS07b]-=-, the authors calculated the second term as (trS)/4, but since S = 2P − , we have trS = 2 dim V −dim V max = 2(dimV −|E|). The last term in the trace formula comes from an combinatorial expansion. Nic... |

1 |
graphs: an introduction and a brief survey, Analysison Graphs and its Applications
- Quantum
(Show Context)
Citation Context ...th a vertex space V associated to G (i.e., a local subspace of V max , see Definition 2.8). In particular, a quantum graph is fixed by the data (V, E, ∂, ℓ, V ). Note that in the literature (see e.g. =-=[Ku08]-=-), a quantum graph is sometimes defined as a metric graph together with a self-adjoint (pseudo-)differential operator acting on it. This definition is more general, since we only associate the Laplaci... |

1 | quantum graphs and boundary triples, to appear - Equilateral - 2007 |

1 | analysis of graphs and related spaces, Habilitation thesis in preparation - Spectral - 2009 |