arXiv:2204.05034

Perfect state transfer in neighborhood coronas

Xing-Kun Song

Published Apr 11, 2022• 0 citations

Abstract

The neighborhood corona $G \star H$ is the graph obtained by taking one copy $G$ and $|G|$ copies of $H$, and joining each vertex of the $j$th copy of $H$ to all neighbors of $v_{j}$ in $G$. In this paper, we study the state transfer of neighborhood coronas related to the adjacency matrix. Concretely, we provide some necessary conditions under which the neighborhood corona $G \star H$ admits perfect state transfer, and obtain a new family of graphs with pretty good state transfer based on neighborhood coronas.

DOI: 10.48550/arxiv.2204.05034

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arXiv:2204.05034

Perfect state transfer in neighborhood coronas

Xing-Kun Song

Published Apr 11, 2022• 0 citations

arXiv HTML PDF

Abstract

The neighborhood corona

G \star H

is the graph obtained by taking one copy

G

and

|G|

copies of

H

, and joining each vertex of the

j

th copy of

H

to all neighbors of

v_{j}

G

. In this paper, we study the state transfer of neighborhood coronas related to the adjacency matrix. Concretely, we provide some necessary conditions under which the neighborhood corona

G \star H

admits perfect state transfer, and obtain a new family of graphs with pretty good state transfer based on neighborhood coronas.

DOI: 10.48550/arxiv.2204.05034

Related papers

Quantum state transfer on edge coronae

Gao-Le Dai, Shu‐Yu Cui, Gui‐Xian Tian +1 more

The edge corona product of a graph $G$ with size $m$ and another graph $H$ , usually expressed as $G\diamond H$ , is formed by taking one copy of $G$ and $m$ copies of $H$ , and then connecting each endpoint of the $i$ -th edge of $G$ to each vertex of the $i$ -th copy of $H$ . This paper investigates perfect state transfer (PST) and pretty good state transfer (PGST) in continuous-time quantum walks on edge corona product of graphs. We establish some sufficient conditions under which this product has no periodic vertices; consequently, PST does not occur. We also exhibit an infinite family of periodic edge corona graphs that do not admit PST, and further give sufficient conditions for the occurrence of PGST on edge corona product of graphs.The edge corona product of a graph $G$ with size $m$ and another graph $H$, usually expressed as $G\diamond H$, is formed by taking one copy of $G$ and $m$ copies of $H$, and then connecting each endpoint of the $i$-th edge of $G$ to each vertex of the $i$-th copy of $H$. This paper investigates perfect state transfer (PST) and pretty good state transfer (PGST) in continuous-time quantum walks on edge corona product of graphs. We establish some sufficient conditions under which this product has no periodic vertices; consequently, PST does not occur. We also exhibit an infinite family of periodic edge corona graphs that do not admit PST, and further give sufficient conditions for the occurrence of PGST on edge corona product of graphs.

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Laplacian state transfer in vertex complemented coronas

Qiang Wang, Xiaogang Liu

arXiv:2202.07202Feb 15, 20220 citations

The neighborhood corona

G \star H

is the graph obtained by taking one copy

G

and

|G|

copies of

H

, and joining each vertex of the

j

th copy of

H

to all neighbors of

v_{j}

G

. In this paper, we study the state transfer of neighborhood coronas related to the adjacency matrix. Concretely, we provide some necessary conditions under which the neighborhood corona

G \star H

admits perfect state transfer, and obtain a new family of graphs with pretty good state transfer based on neighborhood coronas.

The edge corona product of a graph

G

with size

m

and another graph

H

, usually expressed as

G\diamond H

, is formed by taking one copy of

G

and

m

copies of

H

, and then connecting each endpoint of the

i

-th edge of

G

to each vertex of the

i

-th copy of

H

. This paper investigates perfect state transfer (PST) and pretty good state transfer (PGST) in continuous-time quantum walks on edge corona product of graphs. We establish some sufficient conditions under which this product has no periodic vertices; consequently, PST does not occur. We also exhibit an infinite family of periodic edge corona graphs that do not admit PST, and further give sufficient conditions for the occurrence of PGST on edge corona product of graphs.

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs

G

and

H

, the Frucht-Harary corona product

G \circ H

is obtained by taking

|G|

copies of the cone

K_{1} + H

and by identifying the conical vertices according to

G

. Our work explores conditions under which the corona

G \circ H

exhibits state transfer. We also describe new families of graphs with state transfer based on the corona product. Some of these constructions provide a generalization of related known results.

Given a graph

G

with vertex set

V(G)=\{v_1,v_2,\ldots,v_{n_1}\}

and a graph

H

of order

n_2

, the vertex complemented corona, denoted by

G\tilde{\circ}{H}

, is the graph produced by copying

H

n_1

times, with the

i

-th copy of

H

corresponding to the vertex

v_i

, and then adding edges between any vertex in

V(G)\setminus\{v_{i}\}

and any vertex of the

i

-th copy of

H

. The present article deals with quantum state transfer of vertex complemented coronae concerning signless Laplacian matrix. Our research investigates conditions in which signless Laplacian perfect state transfer exists or not on vertex complemented coronae. Additionally, we also provide some mild conditions for the class of graphs under consideration that allow signless Laplacian pretty good state transfer.

We study the existence of quantum state transfer in

\mathcal{Q}

-graphs in this paper. The

\mathcal{Q}

-graph of a graph

G

, denoted by

\mathcal{Q}(G)

, is the graph derived from

G

by plugging a new vertex to each edge of

G

and joining two new vertices which lie on adjacent edges of

G

by an edge. We show that, if all eigenvalues of a regular graph

G

are integers, then its

\mathcal{Q}

-graph

\mathcal{Q}(G)

has no perfect state transfer. In contrast, we also prove that the

\mathcal{Q}

-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of

\mathcal{Q}

-graphs having no perfect state transfer, but admitting pretty good state transfer.