On Moore graphs

Damerell, R. M.

doi:10.1017/s0305004100048015

Cited by 154 publications

(86 citation statements)

References 4 publications

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“…Unlike in the undirected case where nontrivial Moore graphs exist (the Petersen graph, the Hoffman-Singleton graph, and-possibly-graph(s) of degree 57 and diameter 2 of order 3250, cf. [12,1,10]), here it turns out that the equality n d;k ¼ M d;k holds only in the trivial cases when d ¼ 1 or k ¼ 1. The corresponding Moore digraphs are directed cycles of length k þ 1 and complete digraphs of order d þ 1.…”

Section: Introductionmentioning

confidence: 94%

Complete characterization of almost Moore digraphs of degree three

et al. 2004

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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 112 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. ß

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Section: Introductionmentioning

confidence: 94%

Complete characterization of almost Moore digraphs of degree three

et al. 2004

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“…Second, two basic building blocks are interconnected by a link between two servers each in one building block without connecting any two switches directly. Although many efforts [13], [14] have been made to study the degree/diameter problem in graph theory, it is still open in the field of DCN.…”

Section: Motivation and Contributionsmentioning

confidence: 99%

Expandable and Cost-Effective Network Structures for Data Centers Using Dual-Port Servers

Guo

Chen

et al. 2013

IEEE Trans. Comput.

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Abstract-A fundamental goal of data center networking is to efficiently interconnect a large number of servers with the low equipment cost. Several server-centric network structures for data centers have been proposed. They, however, are not truly expandable and suffer a low degree of regularity and symmetry. Inspired by the commodity servers in today's data centers that come with dual port, we consider how to build expandable and cost-effective structures without expensive high-end switches and additional hardware on servers except the two NIC ports. In this paper, two such network structures, called HCN and BCN, are designed, both of which are of server degree 2. We also develop the low overhead and robust routing mechanisms for HCN and BCN. Although the server degree is only 2, HCN can be expanded very easily to encompass hundreds of thousands servers with the low diameter and high bisection width. Additionally, HCN offers a high degree of regularity, scalability, and symmetry, which conform to the modular designs of data centers. BCN is the largest known network structure for data centers with the server degree 2 and network diameter 7. Furthermore, BCN has many attractive features, including the low diameter, high bisection width, large number of node-disjoint paths for the one-to-one traffic, and good fault-tolerant ability. Mathematical analysis and comprehensive simulations show that HCN and BCN possess excellent topological properties and are viable network structures for data centers.

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“…However, Bannai and Ito [1] and Damerell [8] showed that a Moore graph with diameter D > 1 and valency k > 2 must have diameter D = 2 (and valency k ∈ {3, 7, 57}), hence the valency k of Γ should be 2. Therefore Γ is a heptagon, which finishes the proof.…”

Section: An Improved Bound For Diameter Threementioning

confidence: 99%

A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

et al. 2008

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General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. a b s t r a c tWe determine a lower bound for the spectral radius of a graph in terms of the number of vertices and the diameter of the graph. For the specific case of graphs with diameter three we give a slightly better bound. We also construct families of graphs with small spectral radius, thus obtaining asymptotic results showing that the bound is of the right order. We also relate these results to the extremal degree/diameter problem.

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On Moore graphs

Cited by 154 publications

References 4 publications

Complete characterization of almost Moore digraphs of degree three

Complete characterization of almost Moore digraphs of degree three

Expandable and Cost-Effective Network Structures for Data Centers Using Dual-Port Servers

A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

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