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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)

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)

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)

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)

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)

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)

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)

Large scale networks have become ubiquitous elements of our society. Modern social networks , supported by communication and travel technology, have grown in size and complexity to unprecedented scales. Computer networks, such as the Internet, have a fundamental impact on commerce, politics and culture. The study of networks is also central in biology,… (More)