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Graphs of order two less than the Moore bound

- Miller, Mirka, Simanjuntak, Rinovia

**Authors:**Miller, Mirka , Simanjuntak, Rinovia**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 13 (2008), p. 2810-2821**Full Text:**false**Reviewed:****Description:**The Moore bound for a directed graph of maximum out-degree d and diameter k is Md,k=1+d+d2++dk. It is known that digraphs of order Md,k (Moore digraphs) do not exist for d>1 and k>1. Similarly, the Moore bound for an undirected graph of maximum degree d and diameter k is . Undirected Moore graphs only exist in a small number of cases. Mixed (or partially directed) Moore graphs generalize both undirected and directed Moore graphs. In this paper, we shall show that all known mixed Moore graphs of diameter k=2 are unique and that mixed Moore graphs of diameter k3 do not exist.**Description:**C1

Complete characterization of almost moore digraphs of degree three

- Baskoro, Edy, Miller, Mirka, Siran, Jozef, Sutton, Martin

**Authors:**Baskoro, Edy , Miller, Mirka , Siran, Jozef , Sutton, Martin**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 48, no. 2 (2005), p. 112-126**Full Text:**false**Reviewed:****Description:**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. In this paper, we prove that digraphs of degree three and diameter k ≥ 3 which miss the Moore bound by one do not exist. © 2004 Wiley Periodicals, Inc.**Description:**C1**Description:**2003000904

Enumerations of vertex orders of almost Moore digraphs with selfrepeats

- Baskoro, Edy, Cholily, Yus Mochamad, Miller, Mirka

**Authors:**Baskoro, Edy , Cholily, Yus Mochamad , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 1 (2008), p. 123-128**Full Text:**false**Reviewed:****Description:**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 rp (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. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

A sum labelling for the generalised friendship graph

- Fernau, Henning, Ryan, Joe, Sugeng, Kiki Ariyanti

**Authors:**Fernau, Henning , Ryan, Joe , Sugeng, Kiki Ariyanti**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 5-6 (2008), p. 734-740**Full Text:**false**Reviewed:****Description:**We provide an optimal sum labelling scheme for the generalised friendship graph, also known as the flower (a symmetric collection of cycles meeting at a common vertex) and show that its sum number is 2. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

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