Английская Википедия:Degree diameter problem

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In graph theory, the degree diameter problem is the problem of finding the largest possible graph Шаблон:Mvar (in terms of the size of its vertex set Шаблон:Mvar) of diameter Шаблон:Mvar such that the largest degree of any of the vertices in Шаблон:Mvar is at most Шаблон:Mvar. The size of Шаблон:Mvar is bounded above by the Moore bound; for Шаблон:Math and Шаблон:Math only the Petersen graph, the Hoffman-Singleton graph, and possibly graphs (not yet proven to exist) of diameter Шаблон:Math and degree Шаблон:Math attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound.

Formula

Let <math>n_{d,k}</math> be the maximum possible number of vertices for a graph with degree at most d and diameter k. Then <math>n_{d,k}\leq M_{d,k}</math>, where <math>M_{d,k}</math> is the Moore bound:

<math>M_{d,k}=\begin{cases}1+d\frac{(d-1)^k-1}{d-2}&\text{ if }d>2\\2k+1&\text{ if }d=2\end{cases}</math>

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 <math>M_{d,k}=d^k+O(d^{k-1})</math>.

Define the parameter <math>\mu_k=\liminf_{d\to\infty}\frac{n_{d,k}}{d^k}</math>. It is conjectured that <math>\mu_k=1</math> for all k. It is known that <math>\mu_1=\mu_2=\mu_3=\mu_5=1</math> and that <math>\mu_4\geq 1/4</math>.

See also

References


Шаблон:Hyperbolic-geometry-stub Шаблон:Graph-stub

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