Uncertainty principle for discrete Schr\"odinger evolution on graphs

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 $L(\cdot,\cdot)$