Spectral analysis and best decay rate of the wave propagator on the tadpole graph

We consider the damped wave semigroup on the tadpole graph ℛ. We first give a meticulous spectral analysis, followed by a judicious decomposition of the resolvent's kernel. As a consequence, and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of the energy space ℋ, we establish the exponential decay of the corresponding energy, with the optimal decay rate dictated by the spectral abscissa of the relevant operator.