# 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 one more graph (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 be the maximum possible number of vertices for a graph with degree at most *d* and diameter *k*. Then , where is the Moore bound:

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 .

Define the parameter . It is conjectured that for all *k*. It is known that and that . For the general case it is known that . Thus, although it is conjectured that , it is still an open question whether it is in fact exponential.

## See also

## 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*, Mathematical Association of America,**75**(1): 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