Longest Induced Cycles in Circulant Graphs

Fuchs, Elena

doi:10.37236/1949

Cited by 44 publications

(39 citation statements)

References 3 publications

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“…In 2007 Klotz and Sander determined the chromatic number, clique number, independence number, and diameter of X Z/nZ [10]. Other properties of unitary Cayley graphs are studied in [1,4,6,13].…”

Section: Introductionmentioning

confidence: 99%

Domination and upper domination of direct product graphs

Defant

Sumun

2018

Discrete Mathematics

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Let X Z/nZ denote the unitary Cayley graph of Z/nZ. We present results on the tightness of the known inequality γ(X Z/nZ ) ≤ γt(X Z/nZ ) ≤ g(n), where γ and γt denote the domination number and total domination number, respectively, and g is the arithmetic function known as Jacobsthal's function. In particular, we construct integers n with arbitrarily many distinct prime factors such that γ(X Z/nZ ) ≤ γt(X Z/nZ ) ≤ g(n) − 1. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.

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Section: Introductionmentioning

confidence: 99%

Domination and upper domination of direct product graphs

Defant

Sumun

2018

Discrete Mathematics

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“…Various other properties of unitary Cayley graphs were recently investigated. For example, Berrizbeitia and Giudici [6] and Fuchs [9] established the lower and upper bound on the size of the longest induced cycle. Klotz and Sander [13] determined the diameter, clique number, chromatic number and eigenvalues of unitary Cayley graphs.…”

Section: Introductionmentioning

confidence: 99%

Characterization of quantum circulant networks having perfect state transfer

Bašić

2012

Quantum Inf Process

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In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph G, which is characterized by its circulant adjacency matrix A. Formally, we say that there exists a perfect state transfer (PST) between vertices a, b ∈ V (G) if |F (τ ) ab | = 1, for some positive real number τ , where F (t) = exp(ıAt). Saxena, Severini and Shparlinski (International Journal of Quantum Information 5 (2007), 417-430) proved that |F (τ ) aa | = 1 for some a ∈ V (G) and τ ∈ R + if and only if all eigenvalues of G are integer (that is, the graph is integral). The integral circulant graph ICG n (D) has the vertex set Z n = {0, 1, 2, . . . , n − 1} and vertices a and b are adjacent if gcd(a − b, n) ∈ D, where D ⊆ {d : d | n, 1 ≤ d < n}. These graphs are highly symmetric and have important applications in chemical graph theory. We show that ICG n (D) has PST if and only if n ∈ 4N andand a ∈ {1, 2}. We have thus answered the question of complete characterization of perfect state transfer in integral circulant graphs raised in Quantum Information and Computation, Vol. 10, No. 3&4 (2010) 0325-0342 by Angeles-Canul et al. Furthermore, we also calculate perfect quantum communication distance (distance between vertices where PST occurs) and describe the spectra of integral circulant graphs having PST. We conclude by giving a closed form expression calculating the number of integral circulant graphs of a given order having PST.

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“…The gcd-graph X n ({1}) is called the unitary Cayley graph, see [10] and [17] and references therein. In [10], the unitary Cayely graph of a commutative ring R is defined as G R = Cay(R + , U (R)). For more information on G R , we refer the reader to [3], [16] and [18].…”

Section: Introductionmentioning

confidence: 99%

On the Cayley Graph of a Commutative Ring with Respect to its Zero-divisors

Aalipour

Akbari

2016

Communications in Algebra

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Let R be a commutative ring with unity and R + and Z * (R) be the additive group and the set of all non-zero zero-divisors of R, respectively. We denote by CAY(R) the Cayley graph Cay(R + , Z * (R)). In this paper, we study CAY(R). Among other results, it is shown that for every zero-dimensional non-local ring R, CAY(R) is a connected graph of diameter 2. Moreover, for a finite ring R, we obtain the vertex connectivity and the edge connectivity of CAY(R). We investigate rings R with perfect CAY(R) as well. We also study Reg(CAY(R)) the induced subgraph on the regular elements of R. This graph gives a family of vertex transitive graphs. We show that if R is a Noetherian ring and Reg(CAY(R)) has no infinite clique, then R is finite. Furthermore, for every finite ring R, the clique number and the chromatic number of Reg(CAY(R)) are determined. * Proof. Parts (i) and (iii) are obvious. Part (ii) follows from Z(R) = m. Part (iv) holds for every Cayley graph of a group. To prove the last part, note that under an automorphism of graph G, any component of G is isomorphically mapped to another component. Since CAY(R) is vertex-transitive, we conclude that the components of CAY(R) are isomorphic and so (v) is proved.Remark 2. CAY(R) is vertex transitive but it is not necessarily edge transitive. To see this, consider CAY(Z 6 ) ∼ = K 2 K 3 which is not edge transitive (see Figure 1).

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Longest Induced Cycles in Circulant Graphs

Cited by 44 publications

References 3 publications

Domination and upper domination of direct product graphs

Domination and upper domination of direct product graphs

Characterization of quantum circulant networks having perfect state transfer

On the Cayley Graph of a Commutative Ring with Respect to its Zero-divisors

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