## 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.