We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in the Bron-Kerbosch algorithm. All the maximal cliques generated are output in a tree-like form. Subsequently, we prove that its worst-case time complexity is O(3 n/3 ) for an n-vertex graph. This is optimal as a function of n, since there exist up to 3 n/3 maximal cliques in an n-vertex graph. The algorithm is also demonstrated to run very fast in practice by computational experiments.

The association constants of ferrocene with cyclodextrins (CyD’s) in an aqueous medium at 25 °C have been determined by solubility measurements of ferrocene in both the absence and presence of CyD. The solubility of ferrocene in pure water was found to be (4.25 ± 0.02) × 10−5 mol dm−3 at 25.0 ± 0.1 °C. The β- and γ-CyD’s form only 1 : 1 complexes with ferrocene. The 1 : 1 association constants were determined to be (1.39 ± 0.21) × 102 (α), (1.65 ± 0.04) × 104 (β), and (9.04 ± 0.11) × 102 dm3 mol−1 (γ). The 2 : 1 (CyD : ferrocene) association constant was (2.36 ± 0.06) × 103 dm3 mol−1 (α). Ferrocene-included CyD precipitates were also prepared in water, and their stability upon drying in air was studied. The sublimation enthalpy of ferrocene in the temperature range of 290.65 to 298.15 K was estimated to be 74.2 ± 1.5 kJ mol−1.

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We theoretically investigate the dynamics of a gas of strongly interacting Rydberg atoms subject to a time-domain Ramsey interferometry protocol. The many-body dynamics is governed by an Ising-type Hamiltonian with long range interactions of tunable strength. We analyze and model the contrast degradation and phase accumulation of the Ramsey signal and identify scaling laws for varying interrogation times, ensemble densities, and ensemble dimensionalities.Comment: 16 pages, 3 figure