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Fractional revival of threshold graphs under Laplacian dynamics Zhang, Xiaohong


Let $X$ be a graph, and denote its Laplacian matrix by $L$. Let $U(t) = e^{itL}$. Then $U(t)$ is a complex symmetric unitary matrix. We say that $X$ admits Laplacian fractional revival between vertices $j$ and $k$ at time $t = t_0$, if $U(t_0)e_j = \alpha e_j + \beta e_k$ for some complex numbers $\alpha$ and $\beta$ with $\beta\neq0$. In the special case where $\alpha=0$, we say there is perfect state transfer between vertices $j$ and $k$ at time $t = t_0$. Assume that a graph $X$ admits Laplacian fractional revival at time $t=t_0$ between vertices $1$ and $2$. We prove that for the spectral decomposition $L=\sum_{r=0}^q\theta_rE_r$ of $L$, for each $r=0,1,\ldots, q$, either $E_re_1=E_re_2$, or $E_re_1=-E_re_2$, depending on whether $e^{it_0\theta_r}$ equals to 1 or not. That is to say, vertices 1 and 2 are strongly cospectral with respect to $L$. We give a characterization of the parameters of threshold graphs that allow for Laplacian fractional revival between two vertices; those graphs can be used to generate more graphs with Laplacian fractional revival. We also characterize threshold graphs that admit Laplacian fractional revival between a subset of more than two vertices.

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