Pretty good state transfer on double stars

@article{Fan2012PrettyGS,
  title={Pretty good state transfer on double stars},
  author={Xiaoxia Fan and Chris D. Godsil},
  journal={arXiv: Combinatorics},
  year={2012}
}
Let A be the adjacency matrix of a graph $X$ and suppose U(t)=exp(itA). We view A as acting on $\cx^{V(X)}$ and take the standard basis of this space to be the vectors $e_u$ for $u$ in $V(X)$. Physicists say that we have perfect state transfer from vertex $u$ to $v$ at time $\tau$ if there is a scalar $\gamma$ such that $U(\tau)e_u = \gamma e_v$. (Since $U(t)$ is unitary, $\norm\gamma=1$.) For example, if $X$ is the $d$-cube and $u$ and $v$ are at distance $d$ then we have perfect state… 

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