Diameter 2 Cayley graphs of dihedral groups

Erskine, Grahame

doi:10.1016/j.disc.2015.01.023

Cited by 7 publications

(6 citation statements)

References 5 publications

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“…For any integer m ≥ 3 let c m denote the smallest number of elements in a generating set X of a Cayley graph of a cyclic group C(Z m , X) of diameter 2. The best available general upper bound on c m , which is, of course, the degree of C(Z m , X), is c m ≤ 2⌈ √ m⌉, see [8] for a short proof. Applying this to the cyclic group F * of order m = q − 1 we have the existence of a Cayley graph C(F * , X) of diameter 2 with |X| = c q−1 ≤ 2⌈ √ q − 1⌉.…”

Section: Large Cayley Graphs Of Diameter 3 and Small Degreementioning

confidence: 99%

Asymptotically approaching the Moore bound for diameter three by Cayley graphs

Bachratý

Ŝiagiová

Širáň‬

2019

Journal of Combinatorial Theory, Series B

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The largest order n(d, k) of a graph of maximum degree d and diameter k cannot exceed the Moore bound, which has the form M (d, k) = d k − O(d k−1 ) for d → ∞ and any fixed k. Known results in finite geometries on generalised (k + 1)-gons imply, for k = 2, 3, 5, the existence of an infinite sequence of values of d such that n(d, k) = d k − o(d k ). This shows that for k = 2, 3, 5 the Moore bound can be asymptotically approached in the sense that n(d, k)/M (d, k) → 1 as d → ∞; moreover, no such result is known for any other value of k ≥ 2. The corresponding graphs are, however, far from vertex-transitive, and there appears to be no obvious way to extend them to vertex-transitive graphs giving the same type of asymptotic result.The second and the third author (2012) proved by a direct construction that the Moore bound for diameter k = 2 can be asymptotically approached by Cayley graphs. Subsequently, the first and the third author (2015) showed that the same construction can be derived from generalised triangles with polarity.By a detailed analysis of regular orbits of suitable groups of automorphisms of graphs arising from polarity quotients of incidence graphs of generalised quadrangles with polarity, we prove that for an infinite set of values of d there exist Cayley graphs of degree d, diameter 3, and order d 3 −O(d 2.5 ). The Moore bound for diameter 3 can thus as well be asymptotically approached by Cayley graphs. We also show that this method does not extend to constructing Cayley graphs of diameter 5 from generalised hexagons with polarity.

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Section: Large Cayley Graphs Of Diameter 3 and Small Degreementioning

confidence: 99%

Asymptotically approaching the Moore bound for diameter three by Cayley graphs

Bachratý

Ŝiagiová

Širáň‬

2019

Journal of Combinatorial Theory, Series B

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“…However, to our knowledge, the diameter bounds for arbitrary groups in {G m,n,k } have not been studied. This problem also has connections with the well known degree-diameter problem pertaining to this family of graphs (see [7,9,12]). This is the main motivation behind undertaking such an analysis in this paper.…”

Section: Introductionmentioning

confidence: 97%

Bound on the diameter of metacyclic groups

Rajeevsarathy

Sarkar

2019

J. Algebra Appl.

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Let G m,n,k = Zm ⋉ k Zn be the split metacyclic group, where k is a unit modulo n. We derive an upper bound for the diameter of G m,n,k using an arithmetic parameter called the weight, which depends on n, k, and the order of k. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group. t i=1x ai y bi = x a1+...+at y b1k a 2 +...+a t +...+bt−1k a t +bt .

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“…There are also some important topics of graph theory and group theory in the Cayley graphs of dihedral groups. For instance integrality [8], distance-reqular [9], locally primitive [10], degree diameter problem [4] and edge transitivity [5] of Cay(D 2n , S). In this paper, we aim to give the graph structure of Cay(D 2n , S) for n ≥ 3 and |S| = 1, 2 or 3.…”

Section: Introductionmentioning

confidence: 99%

The structure of Cayley graphs of dihedral groups of Valencies 1, 2 and 3

Al-Kaseasbeh¹,

Erfanian²

2021

Proyecciones (Antofagasta)

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Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

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Diameter 2 Cayley graphs of dihedral groups

Cited by 7 publications

References 5 publications

Asymptotically approaching the Moore bound for diameter three by Cayley graphs

Asymptotically approaching the Moore bound for diameter three by Cayley graphs

Bound on the diameter of metacyclic groups

The structure of Cayley graphs of dihedral groups of Valencies 1, 2 and 3

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