A233288 Number of (3/2,2)-tight graphs with 2n vertices, or kinematic chains with 2n links.

1, 1, 2, 16, 230, 6856, 318162, 19819281, 1535380884

Offset: 1

Author

David Eppstein, Dec 06 2013

Keywords

• nonn
• more

Comments

A 2n-vertex graph is (3/2,2)-sparse if every subgraph with k vertices has at most (3/2)k-2 edges, and (3/2,2)-tight if in addition it has exactly 3n-2 edges; see Lee and Streinu (2008). These graphs represent two-dimensional mechanical systems formed by 2n rigid bodies (links), connected at joints where exactly two links are pinned together and can rotate relative to each other, with the entire system having one degree of freedom and having no rigid subsystems. The vertices of the graph represent links and the edges represent joints.

Examples

			For n=1 the single example (a graph with two vertices and one edge) is represented by familiar mechanical systems including door hinges and pairs of scissors. For n=3 the a(3)=2 solutions are the 6-vertex 7-edge graphs Theta(1,3,3) and Theta(2,2,3), each of which has two degree-three vertices connected by three paths of the given lengths. These correspond respectively to the Watt linkage (two four-bar linkages sharing a pair of adjacent links) and the Stephenson linkage.
		

Links

• Eric A. Butcher and Chris Hartman, Efficient enumeration and hierarchical classification of planar simple-jointed kinematic chains: Application to 12- and 14-bar single degree-of-freedom chains, Mechanism and Machine Theory 30 (2005), 1030-1050.
• Martin Larsson, Nauty Laman plugin
• Audrey Lee and Ileana Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (2008), 1425-1437.
• E. E. Peisakh, Structural analysis of planar jointed mechanisms: Current state and problems, J. Machinery Manufacture and Reliability 37 (2008), 207-212.
• Rajesh P. Sunkari and Linda C. Schmidt, Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm, Mechanism and Machine Theory 41 (2006), 1021-1030. This paper sources the 19819281 value for n=8 but contains a typo for n=7.

Crossrefs

Cf. A227117, A273468, A306420, A328060, A328061, A371901.

Programs

nauty

gensparseg 2*$n -K3/2L2 -u # With Laman plugin; see Larsson link.

Extensions

a(9) from Martin Larsson added by Max Alekseyev, Jan 14 2025