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Below is the table of the vertex numbers for the best-known graphs (as of July 2022) 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 ...
The first number in this sequence, 7, is the degree of the Hoffman–Singleton graph, and the McKay–Miller–Širáň graph of degree seven is the Hoffman–Singleton graph. [2] The same construction can also be applied to degrees d {\displaystyle d} for which ( 2 d + 1 ) / 3 {\displaystyle (2d+1)/3} is a prime power but is 0 or −1 mod 4.
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 ...
Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1). The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; [5] for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant, so isomorphic graphs have the same degree sequence. However, the degree ...
Donald Trump’s first day back in at the White House was one without major action on his biggest economic initiative — tariffs — but he made clear that historic new duties are coming.
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 [105] Does a Moore graph with girth 5 and degree 57 exist? [106]