Английская Википедия:Degree diameter problem
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
- Cage (graph theory)
- Table of the largest known graphs of a given diameter and maximal degree
- Table of vertex-symmetric degree diameter digraphs
- Maximum degree-and-diameter-bounded subgraph problem
References
Шаблон:Hyperbolic-geometry-stub
Шаблон:Graph-stub
