2. Periodic Graphs

Ref: Siam Journal of Applied Algebra and Geometry Vol 1, 2017.
Definition (Construction) Let $(\tilde{G} = (V, E),\Gamma)$ be an infinite $d$periodic graph. Let $\Gamma \simeq \mathbb{Z}^d \subseteq Aut(\tilde{G})$. In general, the quotient graph $\tilde{G} / \Gamma$ is a finite multigraph. Alternatively, some $d$periodic graphs could be obtained by $d$ pairwise identifications of nonedges in a finite simple graph. This construction is a model for atombond structures in crystallography since crystals are realizations of 3periodic graphs.Problem 2.1.
[Ciprian Borcea] Which $d$periodic graphs $(\tilde{G}, \Gamma)$ can be obtained by this construction from some finite simple graph $G$? If so, can $G$ be constructed from the $d$periodic graph? 
Problem 2.2.
[Bernd Schulze] Suppose that $(\tilde{G},\tilde{p})$ is periodic in $\mathbb{R}^2$ (with a fixed lattice $L$), such that there exists an equilibrium stress which is strictly positive on all edges and $(\tilde{G},\tilde{p})$ is infinitesimally rigid. Is it true that $\tilde{G}$ is globally rigid in $\mathbb{R}^2$ for any Lgeneric realization? What if $L$ is flexible? 
Periodic Packings
$\psi$ is the map assigning an element of $\mathbb{Z}^2$ to each edge. $\Gamma$generic radii means that the set of coordinates of the points corresponding to vertexorbit representatives is algebraically independent over $\mathbb{Q}$. $(G,\psi)$ is $(2,3,2)$gaintight if $E=2V2$, $E'\leq 2V'2$ for every subgraph $(V',E')$ of $G$ and $E'\leq 2V'3$ for every balanced subgraph $(V',E')$ of $G$ (with the inhertied gain assigment). (A subgraph $H$ is balanced if for every cycle in $H$ the sum of the gains is equal to the identity.)Problem 2.3.
[Bernd Schulze] Let $G=(V,E)$ be a multigraph with $V>1$ and let $(G,\psi)$ be the $\Gamma$labelled quotient gain graph of the contact graph $\tilde G$ of a periodic packing in $\mathbb{R}^2$ with $\Gamma$generic radii. Is it true that $(G,\psi)$ is $(2,3,2)$gainsparse? Moreover if $E=2V2$, is it true that $\tilde G$ is rigid?
The case $V=1$ and $E$ consists of a single loop is a generic radii packing, hence the requirement that $V>1$.
Cite this as: AimPL: Rigidity and flexibility of microstructures, available at http://aimpl.org/flexmicro.