We consider the damped wave semigroup on the tadpole graph \({\mathscr {R}}\) . 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 \({\mathscr {H}}\) , we establish the exponential decay of the corresponding energy, with the optimal decay rate dictated by the spectral abscissa of the relevant operator.
