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2. Periodic Graphs

    1.     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 non-edges in a finite simple graph. This construction is a model for atom-bond structures in crystallography since crystals are realizations of 3-periodic 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 L-generic 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 vertex-orbit representatives is algebraically independent over $\mathbb{Q}$. $(G,\psi)$ is $(2,3,2)$-gain-tight if $|E|=2|V|-2$, $|E'|\leq 2|V'|-2$ for every subgraph $(V',E')$ of $G$ and $|E'|\leq 2|V'|-3$ 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 $(2,3,2)$-gain-sparse? Moreover if $|E|=2|V|-2$, is it true that $\tilde G$ is rigid?
                  Ross 2015 proved that for a fixed lattice $L$, a 2-dimensional $L$-periodic framework is $L$-generically rigid if and only if it has a spanning $(2,3,2)$-gain-sparse subgraph. Hence the conjecture is the exact analogue of the sticky disk theorem of Connelly, Gortler and Theran 2019.

              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.