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\\2d+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 $M_{d,k}\in d^{k}+O(d^{k-1})$ .

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.