Lower bound for the rank of rigidity matrix of 4-valent graphs under   various connectivity assumptions
  In this paper we study the rank of planar rigidity matrix of 4-valent graphs, both in case of generic realizations and configurations in general position, under various connectivity assumptions on the graphs. For each case considered, we prove a lower bound and provide an example which shows the order of the bound we proved is sharp. This work is closed related to work in \cite{Luo:Rigidity} and answers some questions raised there. 
Shisen Luo
math.CO