# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2019: 0.755

SCImago Journal Rank (SJR) 2019: 0.600

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

Article in press

Authors:

S. Kirkland and X. Zhang

Title:

Fractional revival of threshold graphs under Laplacian dynamics

Source:

Discussiones Mathematicae Graph Theory

Received: 2018-11-09, Revised: 2019-05-02, Accepted: 2019-05-02, https://doi.org/10.7151/dmgt.2267

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.

Keywords:

Laplacian matrix, spectral decomposition, quantum information transfer, fractional revival