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