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Moore graphs and beyond: A survey of the degree/diameter problem
 ELECTRONIC JOURNAL OF COMBINATORICS
, 2013
"... 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 bo ..."
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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 stateoftheart 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 Moorelike bounds for special types of graphs and digraphs, such as vertextransitive, Cayley, planar, bipartite, and many others, on
Large digraphs with small diameter: A voltage assignment approach
"... The theory of lifting voltage digraphs provides a useful tool for constructing large digraphs with given properties from suitable small base digraphs endowed with an assignment of voltages (=elements of a finite group) on arcs. We revisit the degree/diameter problem for digraphs from this new perspe ..."
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Cited by 4 (2 self)
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The theory of lifting voltage digraphs provides a useful tool for constructing large digraphs with given properties from suitable small base digraphs endowed with an assignment of voltages (=elements of a finite group) on arcs. We revisit the degree/diameter problem for digraphs from this new perspective and prove a general upper bound on diameter of a lifted digraph in terms of properties of the base digraph and voltage assignment. In addition, we show that all currently known largest vertextransitive Cayley digraphs for semidirect products of groups can be described by means of a voltage assignment construction using simpler groups. This research was done while J. Plesn'ik and J. Sir'an were visiting the Department of Computer Science of the University of Newcastle NSW Australia, supported by small ARC grant. 1 Introduction One fruitful application of graph theory to communication problems is in the design of interconnection networks, such as parallel computers, switching syst...
A remark on almost Moore digraphs of degree three
, 1997
"... It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions ..."
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Cited by 1 (1 self)
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It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. A particularly interesting necessary condition for the existence of a digraph of degree three and diameter k 3 of order one less than the Moore bound is that the number of its arcs be divisible by k + 1. In this paper we derive a new necessary condition (in terms of cycles of the socalled repeat permutation) for the existence of such digraphs of degree three. As a consequence we obtain that a digraph of degree three and diameter k 3 which misses the Moore bound by one cannot be a Cayley digraph of an Abelian group. Research done while the first and the third author were visiting the Department of Computer Science and Software Engineering of the U...
On the structure of almost Moore digraphs containing selfrepeats
"... Abstract. An almost Moore digraph G of degree d>1, diameter k>1 is a diregular digraph with the number of vertices is 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 ..."
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Abstract. An almost Moore digraph G of degree d>1, diameter k>1 is a diregular digraph with the number of vertices is 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 lenght ≤ 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 then G contains exactly k selfrepeats or none. In this paper, we present the possible vertex orders of an almost digraph containing selfrepeats for d ≥ 4, k ≥ 3. 1