Topological densities of periodic graphs

We propose a new method to calculate topological densities of periodic graphs based on the concept of layer-by-layer growth. Topological density is expressed in terms of metric characteristics: the volume of the fundamental domain and the volume of the growth polytope of the graph. Our method is universal (works for all d-periodic graphs) and is easily automated. As examples, we calculate topological densities of all 20 plane 2-uniform graphs and 14 carbon allotrope modifications.

Isogonal non-crystallographic periodic graphs based on knotted sodalite cages

This work considers non-crystallographic periodic nets obtained from multiple identical copies of an underlying crystallographic net by adding or flipping edges so that the result is connected. Such a structure is called a `ladder' net here because the 1-periodic net shaped like an ordinary (infinite) ladder is a particularly simple example. It is shown how ladder nets with no added edges between layers can be generated from tangled polyhedra. These are simply related to the zeolite nets SOD, LTA and FAU. They are analyzed using new extensions of algorithms in the program Systre that allow unambiguous identification of locally stable ladder nets.

Zeta Functions with Respect to General Coined Quantum Walk of Periodic Graphs

We define a zeta function of a graph by using the time evolution matrix of a general coined quantum walk on it, and give a determinant expression for the zeta function of a finite graph. Furthermore, we present a determinant expression for the zeta function of an (inifinite) periodic graph.