Degree diameter problem

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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 maybe a graph 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[edit]

Let n_{d,k} be the maximum possible 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\\2d+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. For the general case it is known that \mu_k\geq 1.6^k. Thus, although is conjectured that \mu_k=1 is still open if it is actually exponential.

See also[edit]

References[edit]

  • Bannai, E.; Ito, T. (1973), "On Moore graphs", J. Fac. Sci. Univ. Tokyo Ser. A 20: 191–208, MR 0323615