Degree diameter problem

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

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, MR 0323615

• Hoffman, Alan J.; Singleton, Robert R. (1960), "Moore graphs with diameter 2 and 3" (PDF), IBM Journal of Research and Development, 5 (4): 497–504, doi:10.1147/rd.45.0497, MR 0140437

• Singleton, Robert R. (1968), "There is no irregular Moore graph", American Mathematical Monthly, 75 (1), Mathematical Association of America: 42–43, doi:10.2307/2315106, JSTOR 2315106, MR 0225679

• Miller, Mirka; Širáň, Jozef (2005), "Moore graphs and beyond: A survey of the degree/diameter problem", Electronic Journal of Combinatorics, Dynamic survey: DS14

• CombinatoricsWiki - The Degree/Diameter Problem