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 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 $n_{d,k}\leq M_{d,k}$ , 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 $\mu_4\geq 1/4$ . For the general case it is known that $\mu_k\geq 1.6^k$ . Thus, although is conjectured that $\mu_k=1$ is still open if it is actually exponential.