Fractional revival of threshold graphs under Laplacian dynamics
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Date
2020
Authors
Kirkland, S.
Zhang, X.
Journal Title
Journal ISSN
Volume Title
Publisher
Discussiones Mathematicae Graph Theory
Abstract
We consider Laplacian fractional revival between two vertices of a graph $X$. Assume that it occurs at time $\tau$ between vertices 1 and 2. We prove that for the spectral decomposition $L = \sum_{r=0}^q \theta_rE_r$ of the Laplacian matrix $L$ of $X$, for each $r = 0, 1, \ldots , q$, either $E_re_1 = E_re_2$, or $E_re_1 = −E_re_2$, depending on whether $e^{i \tau \theta_r}$ 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 within a subset of more than two vertices. Throughout we rely on techniques from spectral graph theory.
Description
Keywords
Laplacian matrix, Spectral decomposition, Quantum information transfer, Fractional revival