# When does the rigidity matrix of a graph have full row rank?

Intuitive description: In the 2D plane, there are $m$ bars connected by $n$ joints. The length of each bar is fixed. These joints and bars can be viewed as a graph (see the figures below). Denote $s_i$ as the static stress of bar $i$.

1. For some graphs (see figure 1), it is clear all $s_i$ must be zero. Otherwise, the stress applied on each joint is non-zero and the graph cannot be balanced in the plane.
2. For other graphs (see figure 2), some $s_i$ can be non-zero and the graph can be well balanced in the plane.

My question: does the later kind of graphs have a name? Any theoretical discussions on them in the literature?

Mathematical description: I am studying graph rigidity. Denote $$R=\mathrm{blkdiag}(e_1^T,\cdots,e_m^T)(H\otimes I_2)$$ as the rigidity matrix of a graph, where $e_i\in\mathbb{R}^2$ denotes the edge of the graph, $H\in\mathbb{R}^{m\times n}$ is the incidence matrix, and $I_2$ is the 2x2 identity matrix. The left null space of $R$ actually is the space of all non-zero stresses. So mathematically my question can be rephrased as: when is the rigidity matrix of full row rank?

• You could check the book J. Graver, B. Servatius, H. Servatius "Combinatorial rigidity"; see if they have a name for it. I think, your class of linkages consists of those whose configuration space is (locally) smooth and has dimension equal the number of edges. I'd call this class "fully flexible". – Misha Apr 25 '13 at 18:42
• @Misha: Thank you so much for the suggestions. In fact, I just borrowed the book you mentioned. There is one subsections talking about this. But not much useful information. Regarding the "fully flexible", I'm not sure if you are meaning "non-rigid". In fact, for rigid graph (or framework), it is still possible to have no nonzero stresses. For example, in figure 2, if I remove one diagonal linkage, the graph is still rigid, but it cannot have any nonzero stresses. – Shiyu Apr 26 '13 at 2:47
• @Misha: I'm not sure if this kind problem has been discussed in the field of mechanics. – Shiyu Apr 26 '13 at 2:50