Some classes of integral circulant graphs either allowing or not allowing perfect state transfer

Bašić, Milan; Petković, Marko D.

doi:10.1016/j.aml.2009.04.007

Cited by 52 publications

(34 citation statements)

References 9 publications

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“…From a graph-theoretic perspective, the main question is whether there is a spectral characterization of graphs which exhibit perfect state transfer. Strong progress along these lines were given on highly structured graphs by Bernasconi et al [5] for hypercubic graphs and by Bašić and Petković [4] for integral circulants. Nevertheless, a general characterization remains elusive (see Godsil [15]).…”

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confidence: 99%

Perfect State Transfer, Graph Products and Equitable Partitions

Yang

Greenberg

Perez

et al. 2011

Int. J. Quantum Inform.

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We describe new constructions of graphs which exhibit perfect state transfer on continuoustime quantum walks. Our constructions are based on variants of the double cones [8,3,2] and the Cartesian graph products (which includes the n-cube Q n ) [11]. Some of our results include:and H is a circulant with odd eigenvalues, their weak product G × H has perfect state transfer. Also, if H is a regular graph with perfect state transfer at time t H and G is a graph where t H |V H | Spec(G) ⊆ 2Zπ, their lexicographic product G[H] has perfect state transfer. For example, these imply Q 2n × H and G[Q n ] have perfect state transfer, whenever H is any circulant with odd eigenvalues and G is any integral graph, for integer n ≥ 2. These complement constructions of perfect state transfer graphs based on Cartesian products. • The double cone K 2 +G on any connected graph G, has perfect state transfer if the weights of the cone edges are proportional to the Perron eigenvector of G. This generalizes results for double cone on regular graphs studied in [8, 3, 2].• For an infinite family G of regular graphs, there is a circulant connection so the graph K 1 + G • G + K 1 has perfect state transfer. In contrast, no perfect state transfer exists if a complete bipartite connection is used (even in the presence of weights) [2]. Moreover, we show that the cylindrical cone K 1 + G + K n + G + K 1 has no perfect state transfer, for any family G of regular graphs.We also describe a generalization of the path collapsing argument [10,11], which reduces questions about perfect state transfer to simpler (weighted) multigraphs, for graphs with equitable distance partitions. Our proofs exploit elementary spectral properties of the underlying graphs.Keywords: perfect state transfer, quantum walk, graph product, equitable partition.Recently, perfect state transfer in continuous-time quantum walks on graphs has received considerable attention. This is due to its potential applications for the transmission of quantum information over quantum networks. It was originally introduced by Bose [7] in the context of quantum walks on linear spin chains or paths. Another reason for this strong interest is due to the universal property *

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confidence: 99%

Perfect State Transfer, Graph Products and Equitable Partitions

Yang

Greenberg

Perez

et al. 2011

Int. J. Quantum Inform.

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“…Bašić and Ilić [12] calculated the clique number of integral circulant graphs with exactly one and two divisors and also provided the inequality for the general case. This paper extends the results from papers [7,6,5]. First it is proven that there exists an integral circulant graph with n vertices having the PST if and only if 4 | n. Several classes of integral circulant graphs having the PST for different values of n are also found.…”

Section: Introductionmentioning

confidence: 74%

“…Furthermore, it is shown that for odd number of vertices, there is no PST, and that among the class of unitary Cayley graphs (subclass of integral circulant graphs), only K 2 (a complete graph with two nodes) and C 4 (cycle of length four) have a PST. In the recent paper [7], the present authors proved that an integral circulant graph with a square-free number of vertices does not have a PST. Two classes of integral circulant graphs having the PST were also found.…”

Section: Introductionmentioning

confidence: 96%

Further results on the perfect state transfer in integral circulant graphs

Petković

Bašić

2011

Computers & Mathematics with Applications

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a b s t r a c tFor a given graph G, denote by A its adjacency matrix and F (t) = exp(iAt). We say that there exist a perfect state transfer (PST) in G if |F (τ ) ab | = 1, for some vertices a, b and a positive real number τ . Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant graphs (circulant graphs with integer eigenvalues). Some results on this topic have already been obtained by Saxena et al. (2007) [5], Bašić et al. (2009) [6] and Basić & Petković (2009) [7]. In this paper, we show that there exists an integral circulant graph with n vertices having a perfect state transfer if and only if 4 | n.Several classes of integral circulant graphs have been found that have a perfect state transfer for the values of n divisible by 4. Moreover, we prove the nonexistence of a PST for several other classes of integral circulant graphs whose order is divisible by 4. These classes cover the class of graphs where the divisor set contains exactly two elements. The obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer.

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“…The 3-dimensional cube is the only connected cubic integral graph with perfect state transfer [96]. Some other results in this direction have been obtained in [8,9].…”

Section: Integral Graphsmentioning

confidence: 94%

“…Load balancing and multiprocessor interconnection networks, 7. Anti-virus protection versus spread of knowledge, 8. Statistical databases and social networks, 9.…”

Section: Introductionmentioning

confidence: 99%

Graph spectral techniques in computer sciences

Arsić

Cvetković

Simić

et al. 2012

APPL ANAL DISCRETE M

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We give a survey of graph spectral techniques used in computer sciences. The survey consists of a description of particular topics from the theory of graph spectra independently of the areas of Computer science in which they are used. We have described the applications of some important graph eigenvalues (spectral radius, algebraic connectivity, the least eigenvalue etc.), eigenvectors (principal eigenvector, Fiedler eigenvector and other), spectral reconstruction problems, spectra of random graphs, Hoffman polynomial, integral graphs etc. However, for each described spectral technique we indicate the fields in which it is used (e.g. in modelling and searching Internet, in computer vision, pattern recognition, data mining, multiprocessor systems, statistical databases, and in several other areas). We present some novel mathematical results (related to clustering and the Hoffman polynomial) as well.

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Some classes of integral circulant graphs either allowing or not allowing perfect state transfer

Cited by 52 publications

References 9 publications

Perfect State Transfer, Graph Products and Equitable Partitions

Perfect State Transfer, Graph Products and Equitable Partitions

Further results on the perfect state transfer in integral circulant graphs

Graph spectral techniques in computer sciences

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