  The tadpole graph consists of a circle and a half-line attached at a vertex.
We analyze standing waves of the nonlinear Schr\"{o}dinger equation with
quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary
conditions at the vertex. The profile of the standing wave with the frequency
$\omega\in (-\infty,0)$ is characterized as a global minimizer of the quadratic
part of energy constrained to the unit sphere in $L^6$. The set of minimizers
includes the set of ground states of the system, which are the global
minimizers of the energy at constant mass ($L^2$-norm), but it is actually
wider. While ground states exist only for a certain interval of masses, the
standing waves exist for every $\omega \in (-\infty,0)$ and correspond to a
bigger interval of masses. It is shown that there exist critical frequencies
$\omega_0$ and $\omega_1$ such that the standing waves are the ground states
for $\omega \in [\omega_0,0)$, local minimizers of the energy at constant mass
for $\omega \in (\omega_1,\omega_0)$, and saddle points of the energy at
constant mass for $\omega \in (-\infty,\omega_1)$. Proofs make use of both the
variational methods and the analytical theory for differential equations.
