  Let $\Lambda\subset \mathbb{R}^d$ be a domain consisting of several cylinders
attached to a bounded center. One says that $\Lambda$ admits a threshold
resonance if there exists a non-trivial bounded function $u$ solving $-\Delta
u=\nu u$ in $\Lambda$ and vanishing at the boundary, where $\nu$ is the bottom
of the essential spectrum of the Dirichlet Laplacian in $\Lambda$. We derive a
sufficient condition for the absence of threshold resonances in terms of the
Laplacian eigenvalues on the center. The proof is elementary and is based on
the min-max principle. Some two- and three-dimensional examples and
applications to the study of Laplacians on thin networks are discussed.
