Degree diameter problem

Unsolved problem in mathematics

Given two positive integers

d

,

k

, what is the largest graph of diameter

k

such that all vertices have degrees at most

d

?

More unsolved problems in mathematics

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 $\mu _{4}\geq 1/4$ .