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- Charles Delorme, Leif K. Jørgensen, Mirka Miller, Guillermo Pineda-Villavicencio
- Eur. J. Comb.
- 2009

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 ∆ ≥ 2, diameter D ≥ 2 and defect 2 (having 2 vertices less than the Moore bipartite bound). We call such graphs bipartite (∆, D, −2)-graphs. We find that the eigenvalues other than ±∆… (More)

- Charles Delorme, Guillermo Pineda-Villavicencio
- Electr. J. Comb.
- 2010

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by ǫ are called graphs with defect or excess ǫ, respectively. While Moore graphs (graphs with ǫ = 0) and graphs with… (More)

- Charles Delorme, Leif K. Jørgensen, Mirka Miller, Guillermo Pineda-Villavicencio
- Journal of Graph Theory
- 2009

We consider bipartite graphs of degree ∆ ≥ 2, diameter D = 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (∆, 3, −2)-graphs.graph. We also prove several necessary conditions for the existence of bipartite (∆, 3, −2)-graphs. The most general of these conditions is that either ∆ or ∆ − 2 must be a… (More)

The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree ∆ and diameter D. Bipartite graphs of maximum degree ∆, diameter D and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D = 2, 3, 4 and 6, and for D = 3, 4, 6, they have been constructed only… (More)

- Eran Nevo, Guillermo Pineda-Villavicencio, David R. Wood
- J. Comb. Theory, Ser. B
- 2016

- Guillermo Pineda-Villavicencio, David R. Wood
- Electr. J. Comb.
- 2015

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree ∆ and diameter k. For fixed k, the answer is Θ(∆ k). We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Θ(∆ k−1), and for graphs of bounded… (More)

- Guillermo Pineda-Villavicencio, José Gómez, Mirka Miller, Hebert Pérez-Rosés
- Networks
- 2009

In the pursuit of obtaining largest graphs of given maximum degree ∆ and diameter D, many construction techniques have been developed. Compounding of graphs is one such technique. In this paper, by means of the compounding of complete graphs into a Moore bipartite graph of diameter 6, we obtain a family of large graphs of the same diameter. 1 maximum… (More)

- Ramiro Feria-Purón, Guillermo Pineda-Villavicencio
- Discrete Applied Mathematics
- 2012

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers ∆ ≥ 2 and D ≥ 2, find the maximum number N b (∆, D) of vertices in a bipartite graph of maximum degree ∆ and diameter D. In this context, the Moore bipartite bound M b (∆, D) represents an upper bound for N b (∆, D). Bipartite graphs of maximum degree ∆, diameter… (More)

- Mirka Miller, Guillermo Pineda-Villavicencio
- Discrete Applied Mathematics
- 2009

- Anthony Dekker, Hebert Pérez-Rosés, Guillermo Pineda-Villavicencio, Paul A. Watters
- J. Math. Model. Algorithms
- 2012

We introduce the problem of finding the largest subgraph of a given weighted undirected graph (host graph), subject to constraints on the maximum degree and the diameter. We discuss some applications in security, network design and parallel processing, and in connection with the latter we derive some bounds for the order of the largest subgraph in host… (More)