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When can perfect state transfer occur? (English) Zbl 1253.05093
Summary: Let $$X$$ be a graph on $$n$$ vertices with adjacency matrix $$A$$ and let $$H(t)$$ denote the matrix-valued function $$exp(iAt)$$. If $$u$$ and $$v$$ are distinct vertices in $$X$$, we say perfect state transfer from $$u$$ to $$v$$ occurs if there is a time $$\tau$$such that $$|H(\tau ) u,v | = 1$$.
The chief problem is to characterize the cases where perfect state transfer occurs. In this paper, it is shown that if perfect state transfer does occur in a graph, then the square of its spectral radius is either an integer or lies in a quadratic extension of the rationals. From this it is deduced that for any integer $$k$$ there only finitely many graphs with maximum valency $$k$$ on which perfect state transfer occurs. It is also shown that if perfect state transfer from $$u$$ to $$v$$ occurs, then the graphs $$X \;u$$ and $$X \;v$$ are cospectral and any automorphism of $$X$$ that fixes $$u$$ must fix $$v$$ (and conversely).

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 81P45 Quantum information, communication, networks (quantum-theoretic aspects)
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