Degree diameter problem

In graph theory, the degree diameter problem is the problem of finding the largest possible graph G (in terms of the size of its vertex set V) of diameter k such that the largest degree of any of the vertices in G is at most d. 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.

be the maximum possible number of vertices for a graph with degree at most d and diameter k. Then

is the Moore bound: This bound is attained for very few graphs, thus the study moves to how close there exist graphs to the Moore bound.

For asymptotic behaviour note that

Define the parameter

μ

lim inf

It is conjectured that

μ

μ

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When the degree is less than or equal to 2 or the diameter is less than or equal to 1, the problem becomes trivial, solved by the cycle graph and complete graph respectively.