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.

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


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

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