Complete characterization of almost Moore digraphs of degree three

Baskoro, Edy Tri; Miller, Mirka; ?ir�?, Jozef; Sutton, Martin

doi:10.1002/jgt.20042

Cited by 19 publications

(20 citation statements)

References 11 publications

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“…Recall that a Moore graph or digraph is a graph or digraph that meets the Moore bound or directed Moore bound, respectively. Let us refer to the survey [30] and articles [8,[10][11][12][13] for more information pertaining to the Moore graphs.…”

Section: Resultsmentioning

confidence: 99%

Distances of Centroid Sets in a Graph-Based Construction for Information Security Applications

et al. 2015

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The aim of this paper is to prove that, for every balanced digraph, in every incidence semiring over a semifield, each centroid set J of the largest distance also has the largest weight, and the distance of J is equal to its weight. This result is surprising and unexpected, because examples show that distances of arbitrary centroid sets in incidence semirings may be strictly less than their weights. The investigation of the distances of centroid sets in incidence semirings of digraphs has been motivated by the information security applications of centroid sets.

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Section: Resultsmentioning

confidence: 99%

Distances of Centroid Sets in a Graph-Based Construction for Information Security Applications

et al. 2015

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“…Let G be a digraph of maximum outdegree d, diameter k and order M (d, k) − δ, then we say G is a (d, k, −δ)-digraph or alternatively a (d, k)-digraph of defect δ. When δ < M (d, k − 1) we have out-regularity, see [5], whereas in general it is not known if we also have in-regularity. Of special interest is the case δ = 1, and a (d, k, −1)-digraph is also denoted as an almost Moore digraph.…”

Section: Introductionmentioning

confidence: 99%

“…Of special interest is the case δ = 1, and a (d, k, −1)-digraph is also denoted as an almost Moore digraph. Almost Moore digraphs do exist for k = 2 as the line digraphs of K d+1 for any d ≥ 2, see [9], whereas (2, k, [10], [5], [7] and [8]. We do know that almost Moore digraphs are diregular for d > 1 and k > 1, see [11].…”

Section: Introductionmentioning

confidence: 99%

Subdigraphs of almost Moore digraphs induced by fixpoints of an automorphism

Sillasen

2015

ejgta

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The degree/diameter problem for directed graphs is the problem of determining the largest possible order for a digraph with given maximum out-degree d and diameter k. An upper bound is given by the Moore bound M (d, k) = In this paper we will look at the structure of subdigraphs of almost Moore digraphs, which are induced by the vertices fixed by some automorphism ϕ. If the automorphism fixes at least three vertices, we prove that the induced subdigraph is either an almost Moore digraph or a diregularAs it is known that almost Moore digraphs have an automorphism r, these results can help us determine structural properties of almost Moore digraphs, such as how many vertices of each order there are with respect to r. We determine this for d = 4 and d = 5, where we prove that except in some special cases, all vertices will have the same order.

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“…For diameter k 3, it is known that there are no (2, k)-digraphs [13]. Recently, it has been proved that, for d 3, there are no (3, k)-digraphs [4]. Thus, it remains to investigate the existence of (d, k)-digraphs when d 4 and k 3.…”

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Enumerations of vertex orders of almost Moore digraphs with selfrepeats

Baskoro

Cholily

Miller

2008

Discrete Mathematics

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An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices one less than the Moore bound. If G is an almost Moore digraph, then for each vertex u ∈ V (G) there exists a vertex v ∈ V (G), called repeat of u and denoted by r(u) = v, such that there are two walks of length k from u to v. The smallest positive integer p such that the composition r p (u) = u is called the order of u. If the order of u is 1 then u is called a selfrepeat. It is known that if G is an almost Moore digraph of diameter k 3 then G contains exactly k selfrepeats or none. In this paper, we propose an exact formula for the number of all vertex orders in an almost Moore digraph G containing selfrepeats, based on the vertex orders of the out-neighbours of any selfrepeat vertex.

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Complete characterization of almost Moore digraphs of degree three

Cited by 19 publications

References 11 publications

Distances of Centroid Sets in a Graph-Based Construction for Information Security Applications

Distances of Centroid Sets in a Graph-Based Construction for Information Security Applications

Subdigraphs of almost Moore digraphs induced by fixpoints of an automorphism

Enumerations of vertex orders of almost Moore digraphs with selfrepeats

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