In this thesis quantum graphs are investigated. Quantum graphs consist
of a finite or countably infinite set of vertices, a finite or countably
infinite set of edges connecting the vertices, a length function for the
edges and differential expressions on each edge with boundary and matching
conditions at the vertices. In order to treat operators on the graph, we
use the direct sum of Hilbert spaces and linear relations in combination
with the in extension theory of symmetric linear relations well known idea
of boundary triplets. In detail, we consider Kirchhoff extensions and also
point interactions. For this, a regularization technique for boundary
triplets together with intermediate extensions is used. Of special interest
is the situation of arbitrarily small edge lenghts. In this case, we show a
correspondence between certain discrete laplacians and the point
interactions. Thereby, the self adjointness, the semi boundedness and the
spectrum of point interactions is described.