Degree diameter problem

Short description: Finding the largest graph of given diameter and degree

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 $n_{d,k}\le M_{d,k}$, where $M_{d,k}$ is the Moore bound:

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

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 ${\mu }_k=\underset{d\to \mathrm{\infty }}{\mathrm{lim\; inf}}\frac{n_{d,k}}{d^k}$. 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 ${\mu }_4\ge 1/4$.

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

References

• Bannai, E.; Ito, T. (1973), "On Moore graphs", J. Fac. Sci. Univ. Tokyo Ser. A 20: 191–208 

• Hoffman, Alan J.; Singleton, Robert R. (1960), "Moore graphs with diameter 2 and 3", IBM Journal of Research and Development 5 (4): 497–504, doi:10.1147/rd.45.0497, http://www.research.ibm.com/journal/rd/045/ibmrd0405H.pdf 

• Singleton, Robert R. (1968), "There is no irregular Moore graph", American Mathematical Monthly (Mathematical Association of America) 75 (1): 42–43, doi:10.2307/2315106 

• Miller; Širáň, Jozef (2005), "Moore graphs and beyond: A survey of the degree/diameter problem", Electronic Journal of Combinatorics Dynamic survey: DS14, http://www.combinatorics.org/ojs/index.php/eljc/article/download/DS14/pdf 

• CombinatoricsWiki - The Degree/Diameter Problem, http://combinatoricswiki.org/wiki/The_Degree/Diameter_Problem