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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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