Enumerations of vertex orders of almost Moore digraphs with selfrepeats

Baskoro, Edy Tri; Cholily, Yus Mochamad; Miller, Mirka

doi:10.1016/j.disc.2007.03.035

Cited by 14 publications

(13 citation statements)

References 12 publications

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“…Cholily, Baskoro and Uttunggadewa [92] gave some conditions for the existence of almost Moore digraphs containing selfrepeat. The smallest positive integer p such that the composition r p (u) = u is called the order of u. Baskoro, Cholily and Miller [30,31] investigated the number of vertex orders present in an almost Moore digraphs containing selfrepeat. An exact formula for the number of all vertex orders in a graph is given, based on the vertex orders of the outneighbours of any selfrepeat vertex.…”

Section: Inspired By the Technique Of Bridges And Touegmentioning

confidence: 99%

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Miller¹,

Širáň‬²

2013

Electron. J. Combin.

217

224

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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter.General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'.This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

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Section: Inspired By the Technique Of Bridges And Touegmentioning

confidence: 99%

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Miller¹,

Širáň‬²

2013

Electron. J. Combin.

217

224

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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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“…Said in another way, the permutation cycles with respect to some automorphism ϕ of the vertices in N + (u) and N − (v) are the same when u and v are selfrepeats. The following lemma is a more general result than that of [2]. Lemma 2.2.…”

Section: Resultsmentioning

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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Enumerations of vertex orders of almost Moore digraphs with selfrepeats

Cited by 14 publications

References 12 publications

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

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

Subdigraphs of almost Moore digraphs induced by fixpoints of an automorphism

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