Degree diameter problem

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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 [math]\displaystyle{ n_{d,k} }[/math] be the maximum possible number of vertices for a graph with degree at most d and diameter k. Then [math]\displaystyle{ n_{d,k}\leq M_{d,k} }[/math], where [math]\displaystyle{ M_{d,k} }[/math] is the Moore bound:

[math]\displaystyle{ M_{d,k}=\begin{cases}1+d\frac{(d-1)^k-1}{d-2}&\text{ if }d\gt 2\\2k+1&\text{ if }d=2\end{cases} }[/math]

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 [math]\displaystyle{ M_{d,k}=d^k+O(d^{k-1}) }[/math].

Define the parameter [math]\displaystyle{ \mu_k=\liminf_{d\to\infty}\frac{n_{d,k}}{d^k} }[/math]. It is conjectured that [math]\displaystyle{ \mu_k=1 }[/math] for all k. It is known that [math]\displaystyle{ \mu_1=\mu_2=\mu_3=\mu_5=1 }[/math] and that [math]\displaystyle{ \mu_4\geq 1/4 }[/math].

See also

References

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




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