  We consider the Schr\"odinger evolution on graph, i.e. solution to the
equation
$\partial_tu(t,\alpha)=i\sum_{\beta\in\mathcal{A}}L(\alpha,\beta)u(t,\beta)$,
here $\mathcal{A}$ is the set of vertices of the graph and the matrix
$(L(\alpha,\beta))_{\alpha,\beta\in\mathcal{A}}$ describes interaction between
the vertices, in particular two vertices $\alpha$ and $\beta$ are connected if
$L(\alpha,\beta)\neq0$. We assume that the graph has a "web-like" structure,
i.e, it consists of an inner part, formed by a finite number of vertices, and
some threads attach to it. We prove that such solution $u(t,\alpha)$ cannot
decay too fast along one thread at two different times, unless it vanishes at
this thread. We also give a characterization of the dimension of the vector
space formed by all the solutions of
$\partial_tu(t,\alpha)=i\sum_{\beta\in\mathcal{A}}L(\alpha,\beta)u(t,\beta)$
when $\mathcal{A}$ is a finite set, in terms of the number of the different
eigenvalues of the matrix $L(\cdot,\cdot)$
