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Fractional revival of threshold graphs under Laplacian dynamics Zhang, Xiaohong
Description
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.
Item Metadata
Title 
Fractional revival of threshold graphs under Laplacian dynamics

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20190425T11:01

Description 
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.

Extent 
45.0 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: University of Manitoba

Series  
Date Available 
20191023

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0384624

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International