Degree diameter problem - Alchetron, the free social encyclopedia

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 maybe a graph 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.

Formula

Let n d , k be the maximum possible of vertices for a graph with degree at most d and diameter k then n d , k ≤ M d , k , where M d , k is the Moore bound:

M d , k = { 1 + d ( d − 1 ) k − 1 d − 2 if d > 2 2 k + 1 if d = 2

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 M d , k = d k + O ( d k − 1 ) .

Define the parameter μ k = lim inf d → ∞ n d , k d k . It is conjectured that μ k = 1 for all k. It is known that μ 1 = μ 2 = μ 3 = μ 5 = 1 and that μ 4 ≥ 1 / 4 . For the general case it is known that μ k ≥ 1.6 k . Thus, although is conjectured that μ k = 1 is still open if it is actually exponential.

References

Degree diameter problem Wikipedia

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