## GENERALISED DISCRETE LAPLACIANS ON GRAPHS AND THEIR RELATION TO QUANTUM GRAPHS

### BibTeX

@MISC{Post_generaliseddiscrete,

author = {Olaf Post},

title = {GENERALISED DISCRETE LAPLACIANS ON GRAPHS AND THEIR RELATION TO QUANTUM GRAPHS},

year = {}

}

### OpenURL

### Abstract

Abstract. The aim of the present paper is to analyse the spectrum of Laplace operators on graphs. Motivated by the general form of vertex conditions of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs using the theory of boundary triples. In particular, we derive a spectral relation for equilateral metric graphs and index formulas. Moreover, we introduce extended metric graphs occuring naturally as limits of “thick ” graphs, and provide spectral analysis of natural Laplacians on such spaces. 1.

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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