Geodesic
Fractional Revival of Threshold Graphs Under Laplacian Dynamics
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 585-600.

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We consider Laplacian fractional revival between two vertices of a graph X. Assume that it occurs at time τ between vertices 1 and 2. We prove that for the spectral decomposition L=∑_r=0^qθ_rE_r of the Laplacian matrix L of X, for each r = 0, 1, . . ., q, either E_re_1 = E_re_2, or E_re_1 = −E_re_2, depending on whether e^i_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 within a subset of more than two vertices. Throughout we rely on techniques from spectral graph theory.
Keywords: Laplacian matrix, spectral decomposition, quantum information transfer, fractional revival 
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Kirkland, Steve; Zhang, Xiaohong. Fractional Revival of Threshold Graphs Under Laplacian Dynamics. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 585-600. http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a13/

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