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Laman Number


The Laman number of a Laman graph G is the number of realizations of G in C^2 compatible with a general edge labeling, counted up to direct complex isometry (Capco et al. 2018).

Equivalently, the Laman number of G is the degree of the algebraic map f_G obtained from the squared-distance equations for G after quotienting translations and direct rotations.

The minimal Laman number among all Laman graphs on n=6, 7, ..., 12 vertices is 2^(n-2) (Capco et al. 2018). Extremal graphs attaining the largest Laman number for a fixed number of vertices are called maximally Laman graphs.


See also

Laman Graph, Laman's Theorem, Maximally Laman Graph, Rigid Graph

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References

Capco, J.; Gallet, M.; Grasegger, G.; Koutschan, C.; Lubbes, N.; and Schicho, J. "The Number of Realizations of a Laman Graph." SIAM J. Appl. Algebra Geom. 2, 94-125, 2018. https://doi.org/10.1137/17M1118312.Grasegger, G.; Koutschan, C.; and Tsigaridas, E. "Lower Bounds on the Number of Realizations of Rigid Graphs." Exper. Math., 2018. https://doi.org/10.1080/10586458.2018.1437851.

Cite this as:

Weisstein, Eric W. "Laman Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LamanNumber.html

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