Results 1 
5 of
5
Eigenvalue bracketing for discrete and metric graphs
 J. Math. Anal. Appl
"... Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the m ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps. 1.
EQUILATERAL QUANTUM GRAPHS AND BOUNDARY TRIPLES
"... Abstract. The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural generalisation of the discrete Laplace operator on the ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural generalisation of the discrete Laplace operator on the underlying graph. These generalised Laplacians are necessary in order to cover general vertex conditions on the metric graph. In case of the standard (also named “Kirchhoff”) conditions, the discrete operator is the usual combinatorial Laplacian. 1.
SPECTRAL ANALYSIS OF METRIC GRAPHS AND RELATED SPACES
"... Abstract. 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 and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the g ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. 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 and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form 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. Moreover, we describe a connection of metric graphs with manifolds. Finally, we comment on Cheeger’s inequality and trace formulas for metric and discrete (generalised) Laplacians. 1.
GENERALISED DISCRETE LAPLACIANS ON GRAPHS AND THEIR RELATION TO QUANTUM GRAPHS
"... 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 t ..."
Abstract
 Add to MetaCart
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.
SPECTRAL ANALYSIS OF METRIC GRAPHS AND RELATED SPACES OLAF POST
, 712
"... Abstract. 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 and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the g ..."
Abstract
 Add to MetaCart
Abstract. 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 and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form 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. Moreover, we describe a connection of metric graphs with manifolds. Finally, we comment on Cheeger’s inequality and trace formulas for metric and discrete (generalised) Laplacians. 1.