Combinatorial aspects of the Löwenstein avoidance rule. Part II: local and global constraints

Abstract

The determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio; the respective motif defines a periodic factor of the net. This article deals with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets, with an application to the 200 2-periodic nets listed on the RCSR (Reticular Chemistry Structure Resource) site. A companion paper described a calculation technique of independence ratios of finite graphs based on propositional calculus. This paper focuses on criteria for the choice of the translation subgroup and of the transversal. The translation subgroup should be chosen in such a way as to eliminate every cycle in the quotient graph that is shorter than structural cycles, or rings, of the net. Topological constraints provide an upper bound to the independence ratio of 2-periodic nets and mostly enable the determination of the associated factor, thus giving a description of a periodic distribution in saturated solid solutions obeying some avoidance rule.