Šiagiová, Jana and Siran, Jozef
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|DOI (Digital Object Identifier) Link:||http://dx.doi.org/doi:10.1016/j.jctb.2011.07.005|
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The order of a graph of maximum degree d and diameter 2 cannot exceed d 2+1, the Moore bound for diameter two. A combination of known results guarantees the existence of regular graphs of degree d, diameter 2, and order at least d2−2d1.525 for all sufficiently large d, asymptotically approaching the Moore bound. The corresponding graphs, however, tend to have a fairly small or trivial automorphism group and the nature of their construction does not appear to allow for modifications that would result in a higher level of symmetry. The best currently available construction of vertex-transitive graphs of diameter 2 and preassigned degree gives order 8/9 (d + ½)2 for all degrees of the form d=(3q−1)/2 for prime powers q=1 mod 4. In this note we show that for an infinite set of degrees d there exist Cayley, and hence vertex-transitive, graphs of degree d, diameter 2, and order d 2−O(d3/2).
|Item Type:||Journal Article|
|Copyright Holders:||2011 Elsevier|
|Funders:||VEGA Research Grants [1/0280/10 and 1/0781/11], APVV Research Grants [0040-06, 0104-07 and 0223-10], APVV LPP Research Grants [0145-06 and 0203-06]|
|Keywords:||degree; diameter; Moore bound; Cayley graph|
|Academic Unit/Department:||Mathematics, Computing and Technology > Mathematics and Statistics|
|Depositing User:||Jozef Siran|
|Date Deposited:||03 Feb 2012 11:28|
|Last Modified:||01 Dec 2012 23:15|
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