scholarly journals On the Automorphism Group of Integral Circulant Graphs

10.37236/555 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Milan Bašić ◽  
Aleksandar Ilić
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Quantum Spin ◽  
Perfect State ◽  
Circulant Graphs ◽  
Spin Networks ◽  
Chemical Graph Theory ◽  
State Transfer ◽  
Vertex Set ◽  
Unitary Cayley Graph

The integral circulant graph $X_n (D)$ has the vertex set $Z_n = \{0, 1,\ldots$, $n{-}1\}$ and vertices $a$ and $b$ are adjacent, if and only if $\gcd(a{-}b$, $n)\in D$, where $D = \{d_1,d_2, \ldots, d_k\}$ is a set of divisors of $n$. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper, we deal with the automorphism group of integral circulant graphs and investigate a problem proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. We determine the size and the structure of the automorphism group of the unitary Cayley graph $X_n (1)$ and the disconnected graph $X_n (d)$. In addition, based on the generalized formula for the number of common neighbors and the wreath product, we completely characterize the automorphism groups $Aut (X_n (1, p))$ for $n$ being a square-free number and $p$ a prime dividing $n$, and $Aut (X_n (1, p^k))$ for $n$ being a prime power.

scholarly journals Perfect State Transfer in Quantum Spin Networks

2004 ◽  
Vol 92 (18) ◽  
Author(s):  
Matthias Christandl ◽  
Nilanjana Datta ◽  
Artur Ekert ◽  
Andrew J. Landahl
Keyword(s):  
Quantum Spin ◽  
Perfect State ◽  
Spin Networks ◽  
State Transfer ◽  
Perfect State Transfer

scholarly journals Hamiltonian properties on a class of circulant interconnection networks

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 71-85 ◽  
Author(s):  
Milan Basic
Keyword(s):  
Cayley Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Cayley Graphs ◽  
Small Diameter ◽  
Quantum Spin ◽  
Perfect State ◽  
Circulant Graphs ◽  
Hamiltonian Paths ◽  
Unitary Cayley Graph

Classes of circulant graphs play an important role in modeling interconnection networks in parallel and distributed computing. They also find applications in modeling quantum spin networks supporting the perfect state transfer. It has been noticed that unitary Cayley graphs as a class of circulant graphs possess many good properties such as small diameter, mirror symmetry, recursive structure, regularity, etc. and therefore can serve as a model for efficient interconnection networks. In this paper we go a step further and analyze some other characteristics of unitary Cayley graphs important for the modeling of a good interconnection network. We show that all unitary Cayley graphs are hamiltonian. More precisely, every unitary Cayley graph is hamiltonian-laceable (up to one exception for X6) if it is bipartite, and hamiltonianconnected if it is not. We prove this by presenting an explicit construction of hamiltonian paths on Xnm using the hamiltonian paths on Xn and Xm for gcd(n,m) = 1. Moreover, we also prove that every unitary Cayley graph is bipancyclic and every nonbipartite unitary Cayley graph is pancyclic.

scholarly journals Quantum walks on graphs of the ordered Hamming scheme and spin networks

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Hiroshi Miki ◽  
Satoshi Tsujimoto ◽  
Luc Vinet
Keyword(s):  
Magnetic Fields ◽  
Quantum Walks ◽  
Perfect State ◽  
Spin Networks ◽  
Krawtchouk Polynomials ◽  
Energy Eigenstates ◽  
State Transfer ◽  
Perfect State Transfer ◽  
Local Magnetic Fields ◽  
Basis Vectors

It is shown that the hopping of a single excitation on certain triangular spin lattices with non-uniform couplings and local magnetic fields can be described as the projections of quantum walks on graphs of the ordered Hamming scheme of depth 2. For some values of the parameters the models exhibit perfect state transfer between two summits of the lattice. Fractional revival is also observed in some instances. The bivariate Krawtchouk polynomials of the Tratnik type that form the eigenvalue matrices of the ordered Hamming scheme of depth 2 give the overlaps between the energy eigenstates and the occupational basis vectors.

scholarly journals A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

2019 ◽  
pp. 729
Author(s):  
Mahsa Mirzargar
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
The Other ◽  
Power Graph ◽  
Commuting Graph ◽  
Vertex Set ◽  
Delta G ◽  
Nite Group

Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

scholarly journals Perfect state transfer in integral circulant graphs

2009 ◽  
Vol 22 (7) ◽  
pp. 1117-1121 ◽  
Author(s):  
Milan Bašić ◽  
Marko D. Petković ◽  
Dragan Stevanović
Keyword(s):  
Perfect State ◽  
Circulant Graphs ◽  
State Transfer ◽  
Perfect State Transfer

scholarly journals PARAMETERS OF INTEGRAL CIRCULANT GRAPHS AND PERIODIC QUANTUM DYNAMICS

2007 ◽  
Vol 05 (03) ◽  
pp. 417-430 ◽  
Author(s):  
NITIN SAXENA ◽  
SIMONE SEVERINI ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Quantum Dynamics ◽  
Perfect State ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Odd Order ◽  
State Transfer ◽  
Network Topologies ◽  
Perfect State Transfer ◽  
Exact Bounds

The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus on relevant properties of integral circulant graphs. Specifically, we bound the number of vertices of integral circulant graphs in terms of their degree, characterize bipartiteness and give exact bounds for their diameter. Additionally, we prove that circulant graphs with odd order do not allow perfect state transfer.

scholarly journals Invariance of KMS states on graph C*-algebras under classical and quantum symmetry

2021 ◽  
pp. 1-17
Author(s):  
Soumalya Joardar ◽  
Arnab Mandal
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Inverse Temperature ◽  
Circulant Graphs ◽  
Connected Graphs ◽  
Kms States ◽  
C Algebras ◽  
Kms State ◽  
Quantum Symmetry ◽  
Strongly Connected

Abstract We study the invariance of KMS states on graph $C^{\ast }$ -algebras coming from strongly connected and circulant graphs under the classical and quantum symmetry of the graphs. We show that the unique KMS state for strongly connected graphs is invariant under the quantum automorphism group of the graph. For circulant graphs, it is shown that the action of classical and quantum automorphism groups preserves only one of the KMS states occurring at the critical inverse temperature. We also give an example of a graph $C^{\ast }$ -algebra having more than one KMS state such that all of them are invariant under the action of classical automorphism group of the graph, but there is a unique KMS state which is invariant under the action of quantum automorphism group of the graph.

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