On the nonexistence of graphs of diameter 2 and defect 2
- Authors: Miller, Mirka , Nguyen, Minh Hoang , Pineda-Villavicencio, Guillermo
- Date: 2009
- Type: Text , Journal article
- Relation: The Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 71, no. (2009), p. 5-20
- Full Text: false
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- Description: In 1960, Hoffman and Singleton investigated the existence of Moore graphs of diameter 2 (graphs of maximum degree d and d² + 1 vertices), and found that such graphs exist only for d = 2; 3; 7 and possibly 57. In 1980, Erdös et al., using eigenvalue analysis, showed that, with the exception of C4, there are no graphs of diameter 2, maximum degree d and d² vertices. In this paper, we show that graphs of diameter 2, maximum degree d and d² - 1 vertices do not exist for most values of d with d ≥ 6, and conjecture that they do not exist for any d ≥ 6.
- Description: 2003007893
New largest known graphs of diameter 6
- Authors: Pineda-Villavicencio, Guillermo , Gómez, José , Miller, Mirka , Pérez-Rosés, Hebert
- Date: 2009
- Type: Text , Journal article
- Relation: Networks Vol. 53, no. 4 (2009), p. 315-328
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- Description: In the pursuit of obtaining largest graphs of given maximum degree
- Description: 2003007890
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
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- 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
New largest graphs of diameter 6. (Extended Abstract)
- Authors: Pineda-Villavicencio, Guillermo , Gomez, Jose , Miller, Mirka , Pérez-Rosés, Hebert
- Date: 2006
- Type: Text , Journal article
- Relation: Electronic Notes in Discrete Mathematics Vol. 24, no. (2006), p. 153-160
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- Description: In the pursuit of obtaining largest graphs of given degree and diameter, many construction techniques have arisen. Compounding of graphs is one such technique. In this paper, by means of the compounding of complete graphs into the bipartite Moore graph of diameter 6, we obtain two families of (
- Description: C1
Complete catalogue of graphs of maximum degree 3 and defect at most 4
- Authors: Miller, Mirka , Pineda-Villavicencio, Guillermo
- Date: 2009
- Type: Text , Journal article
- Relation: Discrete Applied Mathematics Vol. 157, no. 13 (2009), p. 2983-2996
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- Description: We consider graphs of maximum degree 3, diameter D≥2 and at most 4 vertices less than the Moore bound M3,D, that is, (3,D,−)-graphs for ≤4. We prove the non-existence of (3,D,−4)-graphs for D≥5, completing in this way the catalogue of (3,D,−)-graphs with D≥2 and ≤4. Our results also give an improvement to the upper bound on the largest possible number N3,D of vertices in a graph of maximum degree 3 and diameter D, so that N3,D≤M3,D−6 for D≥5. Copyright Elsevier.
Multipartite Moore digraphs
- Authors: Fiol, M. A. , Gimbert, Joan , Miller, Mirka
- Date: 2006
- Type: Text , Journal article
- Relation: Linear Algebra and Its Applications Vol. 419, no. 1 (2006), p. 234-250
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- Description: We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that every vertex of a given partite set is adjacent to the same number δ of vertices in each of the other independent sets. We determine when a multipartite Moore digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a r-partite Moore digraph with interpartite outdegree δ > 1 and diameter k = 2m are obtained. In the case δ = 1, which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive.
- Description: C1
- Description: 2003002157
Hybrid simulated annealing and genetic algorithm for degree/diameter problem
- Authors: Tang, Jianmin , Miller, Mirka , Lin, Yuqing
- Date: 2005
- Type: Text , Conference paper
- Relation: Paper pesented at Sixteenth Australasian Workshop on Combinatorial Algorithms, AWOCA 2005, Ballarat, Victoria : 18th-21st September 2005 p. 321-331
- Full Text: false
- 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 diameter k. It is known that simulated annealing and genetic algorithm are effective techniques to identify global optimization solutions. This paper describes our attempt to build a Hybrid Simulated Annealing and Genetic Algorithm (HSAGA) that can be used to construct larger digraphs, and displays our preliminary results obtained by HSAGA.
- Description: 2003001438
On graphs of defect at most 2
- Authors: Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo
- Date: 2011
- Type: Text , Journal article
- Relation: Discrete Applied Mathematics Vol. 159, no. 13 (2011), p. 1331-1344
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- Description: In this paper we consider the degree/diameter problem, namely, given natural numbers Δ<2 and D<1, find the maximum number N(Δ,D) of vertices in a graph of maximum degree Δ and diameter D. In this context, the Moore bound M(Δ,D) represents an upper bound for N(Δ,D). Graphs of maximum degree Δ, diameter D and order M(Δ,D), called Moore graphs, have turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree Δ<2, diameter D<1 and order M(Δ,D)- with small >0, that is, (Δ,D,-)-graphs. The parameter is called the defect. Graphs of defect 1 exist only for Δ=2. When >1, (Δ,D,-)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in Feria-Purón and Pineda-Villavicencio (2010) [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a (Δ,D,-2)-graph with Δ<4 and D<4 is 2D. Second, and most important, we prove the non-existence of (Δ,D,-2)-graphs with even Δ<4 and D<4; this outcome, together with a proof on the non-existence of (4,3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-)-graphs with D<2 and 0≤≤2. Such a catalogue is only the second census of (Δ,D,-2)-graphs known at present, the first being that of (3,D,-)-graphs with D<2 and 0≤≤2 Jørgensen (1992) [14]. Other results of this paper include necessary conditions for the existence of (Δ,D,-2)-graphs with odd Δ<5 and D<4, and the non-existence of (Δ,D,-2)-graphs with odd Δ<5 and D<5 such that Δ≡0,2(modD). Finally, we conjecture that there are no (Δ,D,-2)-graphs with Δ<4 and D<4, and comment on some implications of our results for the upper bounds of N(Δ,D). © 2011 Elsevier B.V. All rights reserved.