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Uncertainty principle for discrete Schrödinger evolution on graphs

We consider the Schrödinger evolution on graph, i.e. solution to the equation $\partial_tu(t,α)=i\sum_{β\in\mathcal{A}}L(α,β)u(t,β)$, here $\mathcal{A}$ is the set of vertices of the graph and the matrix $(L(α,β))_{α,β\in\mathcal{A}}$ describes interaction between the vertices, in particular two vertices $α$ and $β$ are connected if $L(α,β)\neq0$. 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 $u(t,α)$ 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 $\partial_tu(t,α)=i\sum_{β\in\mathcal{A}}L(α,β)u(t,β)$ when $\mathcal{A}$ is a finite set, in terms of the number of the different eigenvalues of the matrix $L(\cdot,\cdot)$