New largest known graphs of diameter 6

Pineda-Villavicencio, Guillermo; Gómez, J.; Miller, Mirka; Pérez-Rosés, Hebert

doi:10.1002/net.20269

Cited by 4 publications

(5 citation statements)

References 22 publications

(18 reference statements)

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“…By means of the compounding of complete graphs into a bipartite Moore graph of diameter 6, Gómez, Miller, Pérez-Rosés, and Pineda-Villavicencio [284] obtained a family of large graphs of the same diameter. For maximum degrees ∆ = 5, 6, 9, 12 and 14, the members of this family constitute the current largest known graphs of diameter 6.…”

Section: Star Product and Compoundingmentioning

confidence: 99%

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Miller¹,

Širáň‬²

2013

Electron. J. Combin.

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218

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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter.General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'.This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

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Section: Star Product and Compoundingmentioning

confidence: 99%

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Miller¹,

Širáň‬²

2013

Electron. J. Combin.

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218

224

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“…A judicious choice of the building blocks and operations on them can yield large graphs for certain combinations of ∆ and D. This is the principle that has been followed in [44,81] (with graph compounding), and [67] (with voltage assignment), for instance.…”

Section: The Degree/diameter Problemmentioning

confidence: 99%

“…5), can be defined that way. In the Degree/Diameter Problem, the technique was introduced by Bermond, Delorme and Quisquater [6], and later it has been systematically used by other authors, either alone or in combination with other methods (see, for example, [7,8,15,18,22,23,36,41,44,40,42,45,43,81]). …”

Section: Graph Compoundingmentioning

confidence: 99%

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Algebraic and computer-based methods in the undirected degree/diameter problem - A brief survey

Pérez-Rosés

2014

ejgta

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“…Large graphs of diameter 6. I produce a family of large compound graphs of diameter 6 (see [10]). Several members of this family are currently the largest known graphs for their respective maximum degree.…”

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confidence: 99%

Topology of Interconnection Networks With Given Degree and Diameter

Pineda-Villavicencio¹

2010

Bull. Aust. Math. Soc.

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This thesis deals with the design of optimal interconnection networks. Optimality is interpreted as the largest possible number of nodes in the network, under given constraints on the number of connections attached to a node (degree of the node), and on the length of shortest paths between any two nodes (diameter of the network). Any interconnection network with bidirectional channels can be modelled by an undirected graph referred to as the 'topology' of the network. In graph theory this interpretation of optimality is known as the degree/diameter problem [8]. Under these constraints, there is an upper bound M ,D on the maximum number of nodes that a network of maximum degree and diameter D can have, which is known as the Moore bound [2,8]. For bipartite network topology, the bipartite Moore bound M b ,D is defined in a similar way to the general Moore bound [2,8].I address the degree/diameter problem for both general graphs and bipartite graphs. Interconnection networks modeled by (bipartite) Moore graphs-namely, (bipartite) graphs attaining the (bipartite) Moore bound-are optimal.Studies on Moore graphs and bipartite Moore graphs proved their rarity [1,6,12]. Consequently, research efforts to search for such optimal graphs fall into two main streams. (i) Lowering the upper bound on the number of vertices of a graph of maximum degree and diameter D by proving the nonexistence or otherwise of graphs whose number of vertices is close to the corresponding Moore bounds. (ii) Increasing the lower bound on the number of vertices of a graph of maximum degree and diameter D by constructing largest known graphs of maximum degree and diameter D.

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New largest known graphs of diameter 6

Cited by 4 publications

References 22 publications

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Algebraic and computer-based methods in the undirected degree/diameter problem - A brief survey

Topology of Interconnection Networks With Given Degree and Diameter

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