scholarly journals Some Results for the Periodicity and Perfect State Transfer

10.37236/671 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Changjiang Bu ◽  
Jihong Shen
Keyword(s):  
Adjacency Matrix ◽  
Sufficient Condition ◽  
Perfect State ◽  
Necessary And Sufficient Condition ◽  
State Transfer ◽  
Periodic Graph ◽  
Perfect State Transfer ◽  
Necessary And Sufficient

Let $G$ be a graph with adjacency matrix $A$, let $H(t)=\exp(itA)$. $G$ is called a periodic graph if there exists a time $\tau$ such that $H(\tau)$ is diagonal. If $u$ and $v$ are distinct vertices in $G$, we say that perfect state transfer occurs from $u$ to $v$ if there exists a time $\tau$ such that $|H(\tau)_{u,v}|=1$. A necessary and sufficient condition for $G$ is periodic is given. We give the existence for the perfect state transfer between antipodal vertices in graphs with extreme diameter.

scholarly journals Perfect State Transfer on Cayley Graphs over Dihedral Groups: The Non-Normal Case

10.37236/9184 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Xiwang Cao ◽  
Bocong Chen ◽  
San Ling
Keyword(s):  
Group Algebra ◽  
Quantum Information Processing ◽  
Cayley Graphs ◽  
Sufficient Conditions ◽  
Perfect State ◽  
Dihedral Groups ◽  
Normal Case ◽  
State Transfer ◽  
Perfect State Transfer ◽  
Necessary And Sufficient

Recently,  perfect state transfer (PST for short) on graphs has attracted great attention due to their applications in quantum information processing and quantum computations. Many constructions and results have been established through various graphs. However, most of the graphs previously investigated are abelian Cayley graphs. Necessary and sufficient conditions for Cayley graphs over dihedral groups having perfect state transfer were studied recently. The key idea in that paper is the assumption of the normality of the connection set. In those cases, viewed as an element in a group algebra, the connection set is in the center of the group algebra, which makes the situations just like in the abelian case. In this paper, we study the non-normal case. In this case, the discussion becomes more complicated. Using the representations of the dihedral group $D_n$,  we show that ${\rm Cay}(D_n,S)$ cannot have PST if $n$ is odd. For even integers $n$, it is proved that if ${\rm Cay}(D_n,S)$ has PST, then $S$ is normal.

scholarly journals Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks

2022 ◽  
Vol 12 (2) ◽  
pp. 883
Author(s):  
Yuxin Cui ◽  
Shu Li ◽  
Yunxiao Shan ◽  
Fengqiu Liu
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
The State ◽  
Reconstruction Technique ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Control Networks ◽  
State Transfer ◽  
Necessary And Sufficient

This study focuses on the finite-time set reachability of probabilistic Boolean multiplex control networks (PBMCNs). Firstly, based on the state transfer graph (STG) reconstruction technique, the PBMCNs are extended to random logic dynamical systems. Then, a necessary and sufficient condition for the finite-time set reachability of PBMCNs is obtained. Finally, the obtained results are effectively illustrated by an example.

scholarly journals Periodic Graphs

10.37236/510 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Chris Godsil
Keyword(s):  
Adjacency Matrix ◽  
Infinite Family ◽  
Perfect State ◽  
State Transfer ◽  
Scalar Multiple ◽  
The Matrix ◽  
Vertex Transitive ◽  
Perfect State Transfer ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let $X$ be a graph on $n$ vertices with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer from $u$ to $v$ occurs if there is a time $\tau$ such that $|H(\tau)_{u,v}|=1$. If $u\in V(X)$ and there is a time $\sigma$ such that $|H(\sigma)_{u,u}|=1$, we say $X$ is periodic at $u$ with period $\sigma$. It is not difficult to show that if the ratio of distinct non-zero eigenvalues of $X$ is always rational, then $X$ is periodic. We show that the converse holds, from which it follows that a regular graph is periodic if and only if its eigenvalues are distinct. For a class of graphs $X$ including all vertex-transitive graphs we prove that, if perfect state transfer occurs at time $\tau$, then $H(\tau)$ is a scalar multiple of a permutation matrix of order two with no fixed points. Using certain Hadamard matrices, we construct a new infinite family of graphs on which perfect state transfer occurs.

scholarly journals Pretty good state transfer on 1-sum of star graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 1483-1489
Author(s):  
Hailong Hou ◽  
Rui Gu ◽  
Mengdi Tong
Keyword(s):  
Quantum State ◽  
Complex Number ◽  
Adjacency Matrix ◽  
Perfect State ◽  
Quantum State Transfer ◽  
Positive Real ◽  
Star Graphs ◽  
State Transfer ◽  
Good State ◽  
Perfect State Transfer

AbstractLetAbe the adjacency matrix of a graphGand supposeU(t) = exp(itA). We say that we have perfect state transfer inGfrom the vertexuto the vertexvat timetif there is a scalarγof unit modulus such thatU(t)eu=γ ev. It is known that perfect state transfer is rare. So C.Godsil gave a relaxation of this definition: we say that we have pretty good state transfer fromutovif there exists a complex numberγof unit modulus and, for each positive realϵthere is a timetsuch that ‖U(t)eu–γ ev‖ <ϵ. In this paper, the quantum state transfer on 1-sum of star graphsFk,lis explored. We show that there is no perfect state transfer onFk,l, but there is pretty good state transfer onFk,lif and only ifk=l.

scholarly journals Dynamics of Boundary Graphs

2013 ◽  
Vol 5 (3) ◽  
pp. 447-455
Author(s):  
G. Mariumuthu ◽  
M. S. Saraswathy
Keyword(s):  
Simple Graph ◽  
The Other ◽  
Boundary Vertex ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Periodic Graph ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Boundary Graph

In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them. A vertex v is a boundary vertex of a vertex u if for all The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u. If G is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component. Given a positive integer m, the mth iterated boundary graph of G is defined as A graph G is periodic if for some m. A graph G is said to be an eventually periodic graph if there exist positive integers m and k >0 such that We give the necessary and sufficient condition for a graph to be eventually periodic.  Keywords: Boundary graph; Periodic graph. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v5i3.14866 J. Sci. Res. 5 (3), xxx-xxx (2013) 

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

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