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.
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