On the clique number of integral circulant graphs

Bašić, Milan; Ilić, Aleksandar

doi:10.1016/j.aml.2008.12.004

Cited by 31 publications

(43 citation statements)

References 9 publications

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“…These are a subclass of the important and well-researched class of circulant graphs (see Davis [11]) and play a role in quantum physics [26], [6]. These are a subclass of the important and well-researched class of circulant graphs (see Davis [11]) and play a role in quantum physics [26], [6].…”

Section: Introductionmentioning

confidence: 99%

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The energy of integral circulant graphs with prime power order

Sander

Sander²

2011

Appl Anal Discrete Math

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The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a, b} : a, b ∈ Zn, gcd(a − b, n) ∈ D}. For an integral circulant graph on p s vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed p s and varying divisor sets D.2010 Mathematics Subject Classification. Primary. 05C50, Secondary. 15A18.

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Section: Introductionmentioning

confidence: 99%

“…. [12], [8], [18], [25], [6], [7], [15], [2], [16], [22], [3]). , n − 1} and edge set {{a, b} : a, b ∈ Z n , gcd(a − b, n) ∈ D}.…”

Section: Introductionmentioning

confidence: 99%

The energy of integral circulant graphs with prime power order

Sander

Sander²

2011

Appl Anal Discrete Math

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“…For general N , this conjecture can be proved when D(h) is a singleton ( [19]). For arbitrary GCD-graphs, conjecture 2 fails: consider G 20 ({2, 5}) -this example is from [14]. This GCD-graph has a maximal clique size of 6, thus it seems to provide a counter-example to conjecture 2.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Discrete Sampling: A Graph Theoretic Approach to Orthogonal Interpolation

Siripuram

Wu²,

Osgood

2019

IEEE Trans. Inform. Theory

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We study the problem of finding unitary submatrices of the N × N discrete Fourier transform matrix, in the context of interpolating a discrete bandlimited signal using an orthogonal basis. This problem is related to a diverse set of questions on idempotents on Z N and tiling Z N . In this work, we establish a graph-theoretic approach and connections to the problem of finding maximum cliques. We identify the key properties of these graphs that make the interpolation problem tractable when N is a prime power, and we identify the challenges in generalizing to arbitrary N . Finally, we investigate some connections between graph properties and the spectral-tile direction of the Fuglede conjecture.

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“…Klotz and Sander [11] determined the diameter, clique number, chromatic number and eigenvalues of unitary Cayley graphs. 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].…”

Section: Introductionmentioning

confidence: 99%

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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On the clique number of integral circulant graphs

Cited by 31 publications

References 9 publications

The energy of integral circulant graphs with prime power order

The energy of integral circulant graphs with prime power order

Discrete Sampling: A Graph Theoretic Approach to Orthogonal Interpolation

Further results on the perfect state transfer in integral circulant graphs

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