Assur graphs and marginally jammed packings

Jammed packings of soft frictionless spheres are important example systems for the study of granular media and glasses. Here we make an observation about the graph theory attributes of the contact network. It is known that at the unjamming point the contact network is isostatic. We argue that the unjamming point is not only isostatic but minimally isostatic meaning that the network does not possess any proper isostatic subgraphs. Minimally isostatic graphs are also known as Assur graphs. It had been previously observed that isostatic packing derived networks become globally floppy when any bond is removed and globally over-constrained when some bond is added. This is a hallmark property of Assur graphs. Furthermore, we argue that minimal isostaticity may be a necessary consequence of having a repulsive interaction and the first order like nature of the unjamming transition.