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Below is the table of the vertex numbers for the best-known graphs (as of June 2024) in the undirected degree diameter problem for graphs of degree at most 3 ≤ d ≤ 16 and diameter 2 ≤ k ≤ 10. Only a few of the graphs in this table (marked in bold) are known to be optimal (that is, largest possible).
The size of G is bounded above by the Moore bound; for 1 < k and 2 < d, only the Petersen graph, the Hoffman-Singleton graph, and possibly graphs (not yet proven to exist) of diameter k = 2 and degree d = 57 attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound.
an upper bound on the largest possible number of vertices in any graph with this degree and diameter. Therefore, these graphs solve the degree diameter problem for their parameters. Another equivalent definition of a Moore graph G is that it has girth g = 2 k + 1 and precisely n / g ( m – n + 1) cycles of length g , where n and m are ...
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Retirees that optimize the first two variables will receive the biggest possible benefit at a given claim age. But retirees that want the absolute maximum payout must claim Social Security at age ...
Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory.It is much larger than many other large numbers such as Skewes's number and Moser's number, both of which are in turn much larger than a googolplex.
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Degree diameter problem: given two positive integers ,, what is the largest graph of diameter such that all vertices have degrees at most ? Jørgensen's conjecture that every 6-vertex-connected K 6-minor-free graph is an apex graph [109] Does a Moore graph with girth 5 and degree 57 exist? [110]