Degree diameter problem: Difference between revisions

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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

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 $M_{d,k}\in d^{k}+O(d^{k-1})$ .

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.

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, MR 0225679

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

CombinatoricsWiki - The Degree/Diameter Problem

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 In [[graph theory]], the '''degree diameter problem''' is the problem of finding the largest possible [[graph (mathematics)|graph]] ''G'' (in terms of the size of its [[vertex (graph theory)|vertex]] set ''V'') of [[distance (graph theory)|diameter]] ''k'' such that the largest [[degree (graph theory)|degree]] of any of the vertices in ''G'' is at most ''d''. The size of ''G'' is bounded above by the [[Moore graph|Moore bound]]; for 1&nbsp;<&nbsp;''k'' and 2&nbsp;<&nbsp;''d'' only the [[Petersen graph]], the [[Hoffman-Singleton graph]], and maybe a graph of diameter ''k''&nbsp;=&nbsp;2 and degree ''d''&nbsp;=&nbsp;57 attain the Moore bound. In general the largest degree-diameter graphs are much smaller in size than the Moore bound.
+== Formula ==
+Let <math>n_{d,k}</math> be the maximum possible of vertices for a graph with degree at most ''d'' and diameter ''k'' then <math>n_{d,k}\leq M_{d,k}</math>, where <math>M_{d,k}</math> is the [[Moore bound]]:
+:<math>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}</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>M_{d,k}\in d^k+O(d^{k-1})</math>.
+Define the parameter <math>\mu_k=\liminf_{d\to\infty}\frac{n_{d,k}}{d^k}</math>. It is conjectured that <math>\mu_k=1</math> for all ''k''. It is known that <math>\mu_1=\mu_2=\mu_3=\mu_5=1</math> and that <math>\mu_4\geq 1/4</math>. For the general case it is known that <math>\mu_k\geq 1.6^k</math>. Thus, although is conjectured that <math>\mu_k=1</math> is still open if it is actually exponential.
 ==See also==