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 ${\displaystyle \mu _{4}\geq 1/4}$ .

• 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