Almost Moore digraphs are diregular

Miller, Mirka; Gimbert, Joan; Širáň‬, Jozef; Slamin, .

doi:10.1016/s0012-365x(99)00357-x

Cited by 21 publications

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“…It was proved in [14] that, for any k, an almost Moore ðd; kÞ-digraph must in fact be diregular of degree d (i.e., indegrees and outdegrees of all vertices are equal to d) and its diameter must be equal to k. By the same token, no pair of distinct vertices u; v of G can be connected by two u ! v paths of length at most k À 1; informally, we shall say that no pair of vertices of G are connected by two short paths.…”

Section: Preliminariesmentioning

confidence: 99%

Complete characterization of almost Moore digraphs of degree three

et al. 2004

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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 112 examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k ! 3 which miss the Moore bound by one do not exist. ß

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Section: Preliminariesmentioning

confidence: 99%

Complete characterization of almost Moore digraphs of degree three

et al. 2004

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“…Then, the question of finding for which values of z > 1 and k > 1 we have n(0, z, k) = M (0, z, k) − 1, where M (0, z, k) is known as the (directed) Moore bound , becomes an interesting problem. In this case, any extremal digraph turns out to be z-regular (see [11]). Regular digraphs of degree z > 1, diameter k > 1 and order n = z + • • • + z k are called almost Moore (z, k)-digraphs (or (z, k)-digraphs for short).…”

Section: Introductionmentioning

confidence: 99%

On Mixed Almost Moore Graphs of Diameter Two

López

Miret

2016

Electron. J. Combin.

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Mixed almost Moore graphs appear in the context of the Degree/Diameter problem as a class of extremal mixed graphs, in the sense that their order is one less than the Moore bound for mixed graphs. The problem of their existence has been considered before for directed graphs and undirected ones, but not for the mixed case, which is a kind of generalization. In this paper we give some necessary conditions for the existence of mixed almost Moore graphs of diameter two derived from the factorization in $\mathbb{Q}[x]$ of their characteristic polynomial. In this context, we deal with the irreducibility of $\Phi_i(x^2+x-(r-1))$, where $\Phi_i(x)$ denotes the i-th cyclotomic polynomial.

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“…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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Almost Moore digraphs are diregular

Cited by 21 publications

References 0 publications

Complete characterization of almost Moore digraphs of degree three

Complete characterization of almost Moore digraphs of degree three

On Mixed Almost Moore Graphs of Diameter Two

Subdigraphs of almost Moore digraphs induced by fixpoints of an automorphism

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