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.

An unprecedented “strongly” self-catenated MOF containing inclined catenated honeycomb-like units

A 3D MOF {[Co(bibp)(1,4-chdc)]·2H2O}n (1) (bibp = 4,4′-bis(1-imidazolyl)biphenyl, 1,4-H2chdc = 1,4-cyclohexanedicarboxylic acid) was hydrothermally synthesized. This framework buries an unprecedented self-catenated net with the highest topological density among the 4-coordinated nets. The photo-catalytic and electrochemical activities were investigated.

On the Hausdorff measure of regular ω-languages in Cantor space

Automata, Logic and Semantics International audience This paper deals with the calculation of the Hausdorff measure of regular ω-languages, that is, subsets of the Cantor space definable by finite automata. Using methods for decomposing regular ω-languages into disjoint unions of parts of simple structure we derive two sufficient conditions under which ω-languages with a closure definable by a finite automaton have the same Hausdorff measure as this closure. The first of these condition is related to the homogeneity of the local behaviour of the Hausdorff dimension of the underlying set, and the other with a certain topological density of the set in its closure.