Graphs of order two less than the Moore bound
- 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
HSAGA and its application for the construction of near-Moore digraphs
- Authors: Tang, Jianmin , Miller, Mirka , Lin, Yuqing
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Discrete Algorithms Vol. 6, no. 1 (2008), p. 73-84
- Full Text: false
- Reviewed:
- Description: The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. This paper deals with directed graphs. General upper bounds, called Moore bounds, exist for the largest possible order of such digraphs of maximum degree d and given diameter k. It is known that simulated annealing and genetic algorithm are effective techniques to identify global optimal solutions. This paper describes our attempt to build a Hybrid Simulated Annealing and Genetic Algorithm (HSAGA) that can be used to construct large digraphs. We present our new results obtained by HSAGA, as well as several related open problems. © 2007 Elsevier B.V. All rights reserved.
- Description: C1
Structural properties of graphs of diameter 2 with maximal repeats
- Authors: Nguyen, Minh Hoang , Miller, Mirka
- Date: 2008
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 308, no. 11 (Jun 2008), p. 2337-2341
- Full Text: false
- Reviewed:
- Description: It was shown using eigenvalue analysis by Erdos et al. that with the exception of C-4, there are no graphs of diameter 2, of maximum degree d and of order d(2), that is, one less than the Moore bound. These graphs belong to a class of regular graphs of diameter 2, and having certain interesting structural properties, which will be proved in this paper. (c) 2007 Elsevier B.V. All rights reserved.
- Description: C1
On mixed Moore graphs
- Authors: Nguyen, Minh Hoang , Miller, Mirka , Gimbert, Joan
- Date: 2007
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 307, no. 7-8 (2007), p. 964-970
- Full Text: false
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- Description: The Moore bound for a directed graph of maximum out-degree d and diameter k is M
- Description: C1
- Description: 2003005024
Complete characterization of almost moore digraphs of degree three
- 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