On the rigidity of molecular graphs

The rigidity of squares of graphs in three-space has important applications to the study of flexibility in molecules. The Molecular Conjecture, posed in 1984 by T-S. Tay and W. Whiteley, states that the square G2 of a graph G of minimum degree at least two is rigid if and only if G has six spanning trees which cover each edge of G at most five times. We give a lower bound on the degrees of freedom of G2 in terms of forest covers of G. This provides a self-contained proof that the existence of the above six spanning trees is a necessary condition for the rigidity of G2 . In addition, we prove that the truth of the Molecular Conjecture would imply that our lower bound is tight, and would also imply that a conjecture of Jacobs on `independent' squares is valid.

• Type: Journal
• Year: 2008
• Where: COMBINATORICA
• Authors: Bill Jackson, Tibor Jordán
• Keywords: COMBINATORICA 2008, Important Applications, Molecular Conjecture, Spanning Trees