  We develop a detailed rigorous analysis of edge bifurcations of standing
waves in the nonlinear Schr\"odinger (NLS) equation on a tadpole graph (a ring
attached to a semi-infinite line subject to the Kirchhoff boundary conditions
at the junction). It is shown in the recent work [7] by using explicit Jacobi
elliptic functions that the cubic NLS equation on a tadpole graph admits a rich
structure of standing waves. Among these, there are different branches of
localized waves bifurcating from the edge of the essential spectrum of an
associated Schr\"odinger operator. We show by using a modified Lyapunov-Schmidt
reduction method that the bifur- cation of localized standing waves occurs for
every positive power nonlinearity. We distinguish a primary branch of never
vanishing standing waves bifurcating from the trivial solution and an infinite
sequence of higher branches with oscillating behavior in the ring. The higher
branches bifurcate from the branches of degenerate standing waves with
vanishing tail outside the ring. Moreover, we analyze stability of bifurcating
standing waves. Namely, we show that the primary branch is composed by
orbitally stable standing waves for subcritical power nonlinearities, while all
nontrivial higher branches are linearly unstable near the bifurcation point.
The stability character of the degenerate branches remains inconclusive at the
analytical level, whereas heuristic arguments based on analysis of embedded
eigenvalues of negative Krein signatures support the conjecture of their linear
instability at least near the bifurcation point. Numerical results for the
cubic NLS equation show that this conjecture is valid and that the degenerate
branches become spectrally stable far away from the bifurcation point.
