The Rigidity of Infinite Graphs

A rigidity theory is developed for countably infinite simple graphs in \({\mathbb {R}}^d\). Generalisations are obtained for the Laman combinatorial characterisation of generic infinitesimal rigidity for finite graphs in \({\mathbb {R}}^2\) and Tay’s multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in \({\mathbb {R}}^d\). Analogous results are obtained for the classical non-Euclidean \(\ell ^q\) norms.