  The aim of this work is to demonstrate the effectiveness of the extension
theory of symmetric operators in the investigation of the stability of standing
waves for the nonlinear Schr\"odinger equations with two types of
non-linearities (power and logarithmic) and two types of point interactions
($\delta$- and $\delta'$-) on a star graph. Our approach allows us to overcome
the use of variational techniques in the investigation of the Morse index for
self-adjoint operators with non-standard boundary conditions which appear in
the stability study. We also demonstrate how our method simplifies the proof of
the stability results known for the NLS equation with point interactions on the
line.
