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.

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

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

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

Define the parameter ${\displaystyle \mu _{k}=\liminf _{d\to \infty }{\frac {n_{d,k}}{d^{k}}}}$. It is conjectured that ${\displaystyle \mu _{k}=1}$ for all k. It is known that ${\displaystyle \mu _{1}=\mu _{2}=\mu _{3}=\mu _{5}=1}$ and that ${\displaystyle \mu _{4}\geq 1/4}$.