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
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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
On bipartite graphs of diameter 3 and defect 2
- Authors: Delorme, Charles , Jorgensen, Leif , Miller, Mirka , Pineda-Villavicencio, Guillermo
- Date: 2009
- Type: Text , Journal article
- Relation: Journal of Graph Theory Vol. 61, no. 4 (2009), p. 271-288
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- Description: We consider bipartite graphs of degree A<2, diameter D = 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (â–³,3, -2) -graphs. We prove the uniqueness of the known bipartite (3, 3, -2) -graph and bipartite (4, 3, -2)-graph. We also prove several necessary conditions for the existence of bipartite (â–³,3, -2) - graphs. The most general of these conditions is that either â–³ or â–³-2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when â–³ = 6 and â–³ = 9, we prove the non-existence of the corresponding bipartite (â–³,3,-2)-graphs, thus establishing that there are no bipartite (â–³,3, -2)-graphs, for 5
On bipartite graphs of defect 2
- Authors: Delorme, Charles , Jorgensen, Leif , Miller, Mirka , Pineda-Villavicencio, Guillermo
- Date: 2009
- Type: Text , Journal article
- Relation: European Journal of Combinatorics Vol. 30, no. 4 (2009), p. 798-808
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- Description: It is known that the Moore bipartite bound provides an upper bound on the order of a connected bipartite graph. In this paper we deal with bipartite graphs of maximum degree
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.
On graphs of maximum degree 3 and defect 4
- Authors: Pineda-Villavicencio, Guillermo , Miller, Mirka
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of combinatorial mathematics and combinatorial computing Vol. 65, no. (May 2008), p. 25-31
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- Description: It is well known that apart from the Petersen graph there are no Moore graphs of degree 3. As a cubic graph must have an even number of vertices, there are no graphs of maximum degree 3 and
On large bipartite graphs of diameter 3
- Authors: Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo
- Date: 2013
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 313, no. 4 (2013), p. 381-390
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: We consider the bipartite version of the degree/diameter problem, namely, given natural numbers d≥2 and D≥2, find the maximum number N b(d,D) of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-Purón and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,-4)-graphs (that is, bipartite graphs of order M b(d,D)-4) was carried out. Here we first present some structural properties of bipartite (d,3,-4)-graphs, and later prove that there are no bipartite (7,3,-4)-graphs. This result implies that the known bipartite (7,3,-6)-graph is optimal, and therefore Nb(7,3)=80. We dub this graph the Hafner-Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,-4)-graph, and the non-existence of bipartite (6,3,-4)-graphs. In addition, we discover at least one new largest known bipartite-and also vertex-transitive-graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for Nb(11,3). © 2012 Elsevier B.V. All rights reserved.
- Description: 2003011037
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.