On the rigidity of molecular graphs (Q1046739)

A graph is called rigid if a generic bar-and-joint framework corresponding to it is rigid in the usual (kinematic) sense. The authors study the rigidity of squares of graphs, which are also called molecular graphs because they are used to study the flexibility of molecules. The Molecular Conjecture, posed in 1984 by \textit{T. S. Tay} and \textit{W. Whiteley} [Topologie Struct. 9, 31--38 (1984; Zbl 0541.51021)], states that the square \(G^{2}\) of a graph \(G\) of minimum degree at least two is rigid if and only if \(5G\) contains six edge-disjoint spanning trees. The main result of the paper under review is a lower bound on the degrees of freedom of \(G^{2}\) in terms of forest covers of \(G\). This implies that the existence of the above six spanning trees is a necessary condition for the rigidity of \(G^{2}\).