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State transfer on graphs. (English) Zbl 1232.05123
Summary: If $$X$$ is a graph with adjacency matrix $$A$$, then we define $$H(t)$$ to be the operator $$\exp(A)$$. We say that we have perfect state transfer in $$X$$ from the vertex $$u$$ to the vertex $$v$$ at time $$\tau$$ if the $$uv$$-entry of $$|H(\tau )_{u,v}|=1$$. State transfer has been applied to key distribution in commercial cryptosystems, and it seems likely that other applications will be found. We offer a survey of some of the work on perfect state transfer and related questions. The emphasis is almost entirely on the mathematics.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C90 Applications of graph theory
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