Šiagiová, Jana and Siran, Jozef
Approaching the Moore bound for diameter two by Cayleygraphs.
Journal of Combinatorial Theory, Series B, 102(2) pp. 470–473.
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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).
|Project Funding Details:
|Funded Project Name||Project ID||Funding Body|
|Not Set||Not Set||VEGA Research Grants [1/0280/10 and 1/0781/11]|
|Not Set||Not Set||APVV Research Grants [0040-06, 0104-07 and 0223-10]|
|Not Set||Not Set||APVV LPP Research Grants [0145-06 and 0203-06]|
||degree; diameter; Moore bound; Cayley graph
||Mathematics, Computing and Technology > Mathematics and Statistics
||03 Feb 2012 11:28
||28 Aug 2013 08:34
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