Degree diameter problem

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

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)

When the degree is less than or equal to 2 or the diameter is less than or equal to 1, the problem becomes trivial, solved by the cycle graph and complete graph respectively.

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, https://www.combinatorics.org/ojs/index.php/eljc/article/download/DS14/pdf 
• CombinatoricsWiki - The Degree/Diameter Problem, http://combinatoricswiki.org/wiki/The_Degree/Diameter_Problem