arXiv:2502.14460

Signless Laplacian State Transfer on Vertex Complemented Coronae

Zhu, Ke-Yu, Gui‐Xian Tian, Shu‐Yu Cui

Published Feb 20, 2025• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2502.14460

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arXiv:2502.14460

Signless Laplacian State Transfer on Vertex Complemented Coronae

Zhu, Ke-Yu, Gui‐Xian Tian, Shu‐Yu Cui

Published Feb 20, 2025• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.2502.14460

Related papers

Laplacian state transfer in vertex complemented coronas

Qiang Wang, Xiaogang Liu

arXiv:2202.07202Feb 15, 20220 citations

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.

RAIRO - Operations ResearchSep 29, 20260 citations

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Characterizing which graphs have Laplaican perfect state transfer or Laplacian pretty good state transfer is an interesting and desirable problem. In this paper, we investigate the existence of Laplacian perfect state transfer and Laplacian pretty good state transfer in edge coronas. We give sufficient conditions for edge coronas to not have Laplacian perfect state transfer as well as sufficient conditions for edge coronas to have Laplacian perfect state transfer. Additionally, we also give sufficient conditions for edge coronas to have Laplacian pretty good state transfer.Characterizing which graphs have Laplaican perfect state transfer or Laplacian pretty good state transfer is an interesting and desirable problem. In this paper, we investigate the existence of Laplacian perfect state transfer and Laplacian pretty good state transfer in edge coronas. We give sufficient conditions for edge coronas to not have Laplacian perfect state transfer as well as sufficient conditions for edge coronas to have Laplacian perfect state transfer. Additionally, we also give sufficient conditions for edge coronas to have Laplacian pretty good state transfer.

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Xing-Kun Song

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.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.

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Edge State Transfer

Qiuting Chen

Let G be a graph and let t be a positive real number. Then the evolution of the continuous quantum walk defined on G is described by the transition matrix U(t)=exp(itH).The matrix H is called Hamiltonian. So far the most studied quantum walks are the ones whose Hamiltonians are the adjacency matrices of the underlying graphs and initial states are vertex states e_a, with e_a being the characteristic vector of vertex a. \n \nThis thesis focuses on Laplacian edge state transfer, that is, the quantum walks whose initial states are edge states e_a-e_b and Hamiltonians are the Laplacians of the underlying graphs. So far the research about perfect state transfer only involves vertex states and Laplacian edge state transfer has not been studied before. We extend the known results of perfect vertex state transfer and periodicity of vertex states to Laplacian edge state transfer. \n \nWe prove two useful closure properties for perfect Laplacian edge state transfer. One is that complementation preserves perfect edge state transfer. The other is that if G has perfect Laplacian edge state transfer at time τ and H has perfect Laplacian vertex state transfer also at time τ, then with some mild assumption on the pairs of vertex states and edge states that have perfect state transfer, the Cartesian product G □ H also admits perfect edge state transfer. Those two properties provide us new ways to construct graphs with perfect Laplacian edge state transfer. We also observe one phenomenon that happens in Laplacian edge state transfer which never happens in vertex state transfer: if there is perfect state transfer from e_a-e_b to e_α-e_β and also from e_b-e_c to e_β-e_γ at the same time t in G, then G admits perfect state transfer from e_a-e_c to e_α-e_γ at time t. \n \nWe give characterizations of perfect Laplacian edge state transfer in cycles, paths and complete bipartite graphs K_{2,4n}. We study perfect state transfer and periodicity on edge states with special spectral features. We also consider the case when the unsigned Laplacian is Hamiltonian and initial states are plus states of the form e_a+e_b. In this case, we characterize perfect state transfer in paths, cycles and bipartite graphs. \nWe close this thesis by a list of open questions.Let G be a graph and let t be a positive real number. Then the evolution of the continuous quantum walk defined on G is described by the transition matrix U(t)=exp(itH).The matrix H is called Hamiltonian. So far the most studied quantum walks are the ones whose Hamiltonians are the adjacency matrices of the underlying graphs and initial states are vertex states e_a, with e_a being the characteristic vector of vertex a. \n \nThis thesis focuses on Laplacian edge state transfer, that is, the quantum walks whose initial states are edge states e_a-e_b and Hamiltonians are the Laplacians of the underlying graphs. So far the research about perfect state transfer only involves vertex states and Laplacian edge state transfer has not been studied before. We extend the known results of perfect vertex state transfer and periodicity of vertex states to Laplacian edge state transfer. \n \nWe prove two useful closure properties for perfect Laplacian edge state transfer. One is that complementation preserves perfect edge state transfer. The other is that if G has perfect Laplacian edge state transfer at time τ and H has perfect Laplacian vertex state transfer also at time τ, then with some mild assumption on the pairs of vertex states and edge states that have perfect state transfer, the Cartesian product G □ H also admits perfect edge state transfer. Those two properties provide us new ways to construct graphs with perfect Laplacian edge state transfer. We also observe one phenomenon that happens in Laplacian edge state transfer which never happens in vertex state transfer: if there is perfect state transfer from e_a-e_b to e_α-e_β and also from e_b-e_c to e_β-e_γ at the same time t in G, then G admits perfect state transfer from e_a-e_c to e_α-e_γ at time t. \n \nWe give characterizations of perfect Laplacian edge state transfer in cycles, paths and complete bipartite graphs K_{2,4n}. We study perfect state transfer and periodicity on edge states with special spectral features. We also consider the case when the unsigned Laplacian is Hamiltonian and initial states are plus states of the form e_a+e_b. In this case, we characterize perfect state transfer in paths, cycles and bipartite graphs. \nWe close this thesis by a list of open questions.

Jan 11, 20192 citations

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 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 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.