  We study the nonlinear Schr\"odinger equation (NLS) on a star graph
$\mathcal{G}$. At the vertex an interaction occurs described by a boundary
condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate an
orbital instability of the standing waves $e^{i\omega t}\mathbf{\Phi}(x)$ of
NLS-$\delta$ equation with attractive power nonlinearity on $\mathcal{G}$ when
the profile $\Phi(x)$ has mixed structure (i.e. has bumps and tails). In our
approach we essentially use the extension theory of symmetric operators by
Krein - von Neumann, and the analytic perturbations theory, avoiding the
variational techniques standard in the stability study. We also prove orbital
stability of the unique standing wave solution of NLS-$\delta$ equation with
repulsive nonlinearity.

Spectral Theory;Analysis of PDEs