Degree diameter problem explained

In graph theory, the degree diameter problem is the problem of finding the largest possible graph (in terms of the size of its vertex set) of diameter such that the largest degree of any of the vertices in is at most . The size of is bounded above by the Moore bound; for and, only the Petersen graph, the Hoffman-Singleton graph, and possibly graphs (not yet proven to exist) of diameter and degree 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 ((d -1 ) ^{k-1})/(d -2) & if d> 2 \ 2 k +1 & 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\toinfty} (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 .

See also

• 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